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Generating function

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about generating functions in mathematics. For generating functions in classical mechanics, see <a href="Generating_function_(physics)" title="Generating function (physics)">Generating function (physics)</a>. For generators in computer programming, see <a href="Generator_(computer_programming)" title="Generator (computer programming)">Generator (computer programming)</a>. For the moment generating function in statistics, see <a href="Moment_generating_function" class="mw-redirect" title="Moment generating function">Moment generating function</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, a generating function is a representation of an <a href="Infinite_sequence" class="mw-redirect" title="Infinite sequence">infinite sequence</a> of numbers as the <a href="Coefficient" title="Coefficient">coefficients</a> of a <a href="Formal_power_series" title="Formal power series">formal power series</a>. Generating functions are often expressed in <a href="Closed-form_expression" title="Closed-form expression">closed form</a> (rather than as a series), by some expression involving operations on the formal series.

There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed.

Generating functions are sometimes called generating series,<a href="#cite_note-1">[1]</a> in that a series of terms can be said to be the generator of its sequence of term coefficients.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

Generating functions were first introduced by <a href="Abraham_de_Moivre" title="Abraham de Moivre">Abraham de Moivre</a> in 1730, in order to solve the general linear recurrence problem.<a href="#cite_note-2">[2]</a>

<a href="George_P%C3%B3lya" title="George Pólya">George Pólya</a> writes in <a href="Mathematics_and_plausible_reasoning" class="mw-redirect" title="Mathematics and plausible reasoning">Mathematics and plausible reasoning</a>:

<blockquote>The name "generating function" is due to <a href="Laplace" class="mw-redirect" title="Laplace">Laplace</a>. Yet, without giving it a name, <a href="Euler" class="mw-redirect" title="Euler">Euler</a> used the device of generating functions long before Laplace [..]. He applied this mathematical tool to several problems in Combinatory Analysis and the <a href="Number_theory" title="Number theory">Theory of Numbers</a>.</blockquote>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

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</style><blockquote class="templatequote">A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in a bag, and then we have only one object to carry, the bag.</blockquote><div class="templatequotecite">— <a href="George_P%C3%B3lya" title="George Pólya">George Pólya</a>, <a href="Mathematics_and_plausible_reasoning" class="mw-redirect" title="Mathematics and plausible reasoning">Mathematics and plausible reasoning</a> (1954)</div>

<blockquote class="templatequote">A generating function is a clothesline on which we hang up a sequence of numbers for display.</blockquote><div class="templatequotecite">— <a href="Herbert_Wilf" title="Herbert Wilf">Herbert Wilf</a>, <a rel="nofollow" class="external text" href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a> (1994)</div>

<div class="mw-heading mw-heading3"><h3 id="Convergence">Convergence</h3></div>

Unlike an ordinary series, the formal <a href="Power_series" title="Power series">power series</a> is not required to <a href="Convergent_series" title="Convergent series">converge</a>: in fact, the generating function is not actually regarded as a <a href="Function_(mathematics)" title="Function (mathematics)">function</a>, and the "variable" remains an <a href="Indeterminate_(variable)" class="mw-redirect" title="Indeterminate (variable)">indeterminate</a>. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. Thus generating functions are not functions in the formal sense of a mapping from a <a href="Domain_of_a_function" title="Domain of a function">domain</a> to a <a href="Codomain" title="Codomain">codomain</a>.

These expressions in terms of the indeterminate x may involve arithmetic operations, differentiation with respect to x and composition with (i.e., substitution into) other generating functions; since these operations are also defined for functions, the result looks like a function of x. Indeed, the closed form expression can often be interpreted as a function that can be evaluated at (sufficiently small) concrete values of x, and which has the formal series as its <a href="Series_expansion" title="Series expansion">series expansion</a>; this explains the designation "generating functions". However such interpretation is not required to be possible, because formal series are not required to give a <a href="Convergent_series" title="Convergent series">convergent series</a> when a nonzero numeric value is substituted for x.

<div class="mw-heading mw-heading3"><h3 id="Limitations">Limitations</h3></div>

Not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative and fractional powers of x are examples of functions that do not have a corresponding formal power series.

<div class="mw-heading mw-heading2"><h2 id="Types">Types</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Ordinary_generating_function_(OGF)">Ordinary generating function (OGF)</h3></div>

When the term generating function is used without qualification, it is usually taken to mean an ordinary generating function. The ordinary generating function of a sequence an is:

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<annotation encoding="application/x-tex">{\displaystyle G(a_{n};x)=\sum _{n=0}^{\infty }a_{n}x^{n}.}</annotation>

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If an is the <a href="Probability_mass_function" title="Probability mass function">probability mass function</a> of a <a href="Discrete_random_variable" class="mw-redirect" title="Discrete random variable">discrete random variable</a>, then its ordinary generating function is called a <a href="Probability-generating_function" title="Probability-generating function">probability-generating function</a>.

<div class="mw-heading mw-heading3"><h3 id="Exponential_generating_function_(EGF)">Exponential generating function (EGF)</h3></div>

The exponential generating function of a sequence an is

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {EG} (a_{n};x)=\sum _{n=0}^{\infty }a_{n}{\frac {x^{n}}{n!}}.}</annotation>

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Exponential generating functions are generally more convenient than ordinary generating functions for <a href="Combinatorial_enumeration" class="mw-redirect" title="Combinatorial enumeration">combinatorial enumeration</a> problems that involve labelled objects.<a href="#cite_note-3">[3]</a>

Another benefit of exponential generating functions is that they are useful in transferring linear <a href="Recurrence_relations" class="mw-redirect" title="Recurrence relations">recurrence relations</a> to the realm of <a href="Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a>. For example, take the <a href="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a> {fn} that satisfies the linear recurrence relation fn+2 = fn+1 + fn. The corresponding exponential generating function has the form

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {EF} (x)=\sum _{n=0}^{\infty }{\frac {f_{n}}{n!}}x^{n}}</annotation>

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and its derivatives can readily be shown to satisfy the differential equation EF″(x) = EF′(x) + EF(x) as a direct analogue with the recurrence relation above. In this view, the factorial term n! is merely a counter-term to normalise the derivative operator acting on xn.

<div class="mw-heading mw-heading3"><h3 id="Poisson_generating_function">Poisson generating function</h3></div>

The Poisson generating function of a sequence an is

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {PG} (a_{n};x)=\sum _{n=0}^{\infty }a_{n}e^{-x}{\frac {x^{n}}{n!}}=e^{-x}\,\operatorname {EG} (a_{n};x).}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Lambert_series">Lambert series</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Lambert_series" title="Lambert series">Lambert series</a></div>

The Lambert series of a sequence an is

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {LG} (a_{n};x)=\sum _{n=1}^{\infty }a_{n}{\frac {x^{n}}{1-x^{n}}}.}</annotation>

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</math>Note that in a Lambert series the index n starts at 1, not at 0, as the first term would otherwise be undefined.

The Lambert series coefficients in the power series expansions

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<annotation encoding="application/x-tex">{\displaystyle b_{n}:=[x^{n}]\operatorname {LG} (a_{n};x)}</annotation>

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</math>for integers n ≥ 1 are related by the <a href="Divisor_sum_identities" title="Divisor sum identities">divisor sum</a>

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<annotation encoding="application/x-tex">{\displaystyle b_{n}=\sum _{d|n}a_{d}.}</annotation>

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</math>The <a href="Lambert_series" title="Lambert series">main article</a> provides several more classical, or at least well-known examples related to special <a href="Arithmetic_functions" class="mw-redirect" title="Arithmetic functions">arithmetic functions</a> in <a href="Number_theory" title="Number theory">number theory</a>. As an example of a Lambert series identity not given in the main article, we can show that for |x|, |xq| < 1 we have that <a href="#cite_note-4">[4]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {q^{n}x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }{\frac {q^{n}x^{n^{2}}}{1-qx^{n}}}+\sum _{n=1}^{\infty }{\frac {q^{n}x^{n(n+1)}}{1-x^{n}}},}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {q^{n}x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }{\frac {q^{n}x^{n^{2}}}{1-qx^{n}}}+\sum _{n=1}^{\infty }{\frac {q^{n}x^{n(n+1)}}{1-x^{n}}},}</annotation>

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where we have the special case identity for the generating function of the <a href="Divisor_function" title="Divisor function">divisor function</a>, d(n) ≡ σ0(n), given by<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }{\frac {x^{n^{2}}\left(1+x^{n}\right)}{1-x^{n}}}.}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {x^{n}}{1-x^{n}}}=\sum _{n=1}^{\infty }{\frac {x^{n^{2}}\left(1+x^{n}\right)}{1-x^{n}}}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Bell_series">Bell series</h3></div>

The <a href="Bell_series" title="Bell series">Bell series</a> of a sequence an is an expression in terms of both an indeterminate x and a prime p and is given by:<a href="#cite_note-5">[5]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {BG} _{p}(a_{n};x)=\sum _{n=0}^{\infty }a_{p^{n}}x^{n}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Dirichlet_series_generating_functions_(DGFs)">Dirichlet series generating functions (DGFs)</h3></div>

<a href="Formal_Dirichlet_series" class="mw-redirect" title="Formal Dirichlet series">Formal Dirichlet series</a> are often classified as generating functions, although they are not strictly formal power series. The Dirichlet series generating function of a sequence an is:<a href="#cite_note-W56-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {DG} (a_{n};s)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n^{s}}}.}</annotation>

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The Dirichlet series generating function is especially useful when an is a <a href="Multiplicative_function" title="Multiplicative function">multiplicative function</a>, in which case it has an <a href="Euler_product" title="Euler product">Euler product</a> expression<a href="#cite_note-W59-7">[7]</a> in terms of the function's Bell series:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {DG} (a_{n};s)=\prod _{p}\operatorname {BG} _{p}(a_{n};p^{-s})\,.}</annotation>

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If an is a <a href="Dirichlet_character" title="Dirichlet character">Dirichlet character</a> then its Dirichlet series generating function is called a <a href="Dirichlet_L-series" class="mw-redirect" title="Dirichlet L-series">Dirichlet L-series</a>. We also have a relation between the pair of coefficients in the <a href="Lambert_series" title="Lambert series">Lambert series</a> expansions above and their DGFs. Namely, we can prove that:

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<annotation encoding="application/x-tex">{\displaystyle [x^{n}]\operatorname {LG} (a_{n};x)=b_{n}}</annotation>

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</math>if and only if

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {DG} (a_{n};s)\zeta (s)=\operatorname {DG} (b_{n};s),}</annotation>

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</math>where ζ(s) is the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>.<a href="#cite_note-8">[8]</a>

The sequence ak generated by a <a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a> generating function (DGF) corresponding to:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {DG} (a_{k};s)=\zeta (s)^{m}}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {DG} (a_{k};s)=\zeta (s)^{m}}</annotation>

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</math>has the ordinary generating function:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=1}^{k=n}a_{k}x^{k}=x+{\binom {m}{1}}\sum _{2\leq a\leq n}x^{a}+{\binom {m}{2}}{\underset {ab\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }}}x^{ab}+{\binom {m}{3}}{\underset {abc\leq n}{\sum _{a=2}^{\infty }\sum _{c=2}^{\infty }\sum _{b=2}^{\infty }}}x^{abc}+{\binom {m}{4}}{\underset {abcd\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }\sum _{c=2}^{\infty }\sum _{d=2}^{\infty }}}x^{abcd}+\cdots }">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{k=n}a_{k}x^{k}=x+{\binom {m}{1}}\sum _{2\leq a\leq n}x^{a}+{\binom {m}{2}}{\underset {ab\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }}}x^{ab}+{\binom {m}{3}}{\underset {abc\leq n}{\sum _{a=2}^{\infty }\sum _{c=2}^{\infty }\sum _{b=2}^{\infty }}}x^{abc}+{\binom {m}{4}}{\underset {abcd\leq n}{\sum _{a=2}^{\infty }\sum _{b=2}^{\infty }\sum _{c=2}^{\infty }\sum _{d=2}^{\infty }}}x^{abcd}+\cdots }</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Polynomial_sequence_generating_functions">Polynomial sequence generating functions</h3></div>

The idea of generating functions can be extended to sequences of other objects. Thus, for example, polynomial sequences of <a href="Binomial_type" title="Binomial type">binomial type</a> are generated by:

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<annotation encoding="application/x-tex">{\displaystyle e^{xf(t)}=\sum _{n=0}^{\infty }{\frac {p_{n}(x)}{n!}}t^{n}}</annotation>

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</math>where pn(x) is a sequence of polynomials and f(t) is a function of a certain form. <a href="Sheffer_sequence" title="Sheffer sequence">Sheffer sequences</a> are generated in a similar way. See the main article <a href="Generalized_Appell_polynomials" title="Generalized Appell polynomials">generalized Appell polynomials</a> for more information.

Examples of <a href="Polynomial_sequence" title="Polynomial sequence">polynomial sequences</a> generated by more complex generating functions include:

<ul><li><a href="Appell_polynomials" class="mw-redirect" title="Appell polynomials">Appell polynomials</a></li>

<li><a href="Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a></li>

<li><a href="Difference_polynomials" title="Difference polynomials">Difference polynomials</a></li>

<li><a href="Generalized_Appell_polynomials" title="Generalized Appell polynomials">Generalized Appell polynomials</a></li>

<li><a href="Q-difference_polynomial" title="Q-difference polynomial">q-difference polynomials</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Other_generating_functions">Other generating functions</h3></div>

Other sequences generated by more complex generating functions include:

<ul><li>Double exponential generating functions e.g. the <a href="Bell_numbers" class="mw-redirect" title="Bell numbers">Bell numbers</a></li>

<li>Hadamard products of generating functions and diagonal generating functions, and their corresponding <a href="Generating_function_transformation#Hadamard_products_and_diagonal_generating_functions" title="Generating function transformation">integral transformations</a></li></ul>

<div class="mw-heading mw-heading4"><h4 id="Convolution_polynomials">Convolution polynomials</h4></div>

Knuth's article titled "Convolution Polynomials"<a href="#cite_note-9">[9]</a> defines a generalized class of convolution polynomial sequences by their special generating functions of the form

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<annotation encoding="application/x-tex">{\displaystyle F(z)^{x}=\exp {\bigl (}x\log F(z){\bigr )}=\sum _{n=0}^{\infty }f_{n}(x)z^{n},}</annotation>

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for some analytic function F with a power series expansion such that F(0) = 1.

We say that a family of polynomials, f0, f1, f2, ..., forms a convolution family if <a href="Degree_of_a_polynomial" title="Degree of a polynomial">deg</a> fn ≤ n and if the following convolution condition holds for all x, y and for all n ≥ 0:

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<annotation encoding="application/x-tex">{\displaystyle f_{n}(x+y)=f_{n}(x)f_{0}(y)+f_{n-1}(x)f_{1}(y)+\cdots +f_{1}(x)f_{n-1}(y)+f_{0}(x)f_{n}(y).}</annotation>

</semantics>

</math>

We see that for non-identically zero convolution families, this definition is equivalent to requiring that the sequence have an ordinary generating function of the first form given above.

A sequence of convolution polynomials defined in the notation above has the following properties:

<ul><li>The sequence n! · fn(x) is of <a href="Binomial_type" title="Binomial type">binomial type</a></li>

<li>Special values of the sequence include fn(1) = [zn] F(z) and fn(0) = δn,0, and</li>

<li>For arbitrary (fixed) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,t\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle x,y,t\in \mathbb {C} }</annotation>

</semantics>

</math><img src="./cb2604c2e32ced47b462417730dab22fa26a2c3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.509ex;" alt="{\displaystyle x,y,t\in \mathbb {C} }" loading="lazy">, these polynomials satisfy convolution formulas of the form</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f_{n}(x+y)&=\sum _{k=0}^{n}f_{k}(x)f_{n-k}(y)\\f_{n}(2x)&=\sum _{k=0}^{n}f_{k}(x)f_{n-k}(x)\\xnf_{n}(x+y)&=(x+y)\sum _{k=0}^{n}kf_{k}(x)f_{n-k}(y)\\{\frac {(x+y)f_{n}(x+y+tn)}{x+y+tn}}&=\sum _{k=0}^{n}{\frac {xf_{k}(x+tk)}{x+tk}}{\frac {yf_{n-k}(y+t(n-k))}{y+t(n-k)}}.\end{aligned}}}</annotation>

</semantics>

</math>

For a fixed non-zero parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\in \mathbb {C} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle t\in \mathbb {C} }</annotation>

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</math><img src="./d53146084bd2654adeb7cb529c2e96ad39e6b62f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.358ex; height:2.176ex;" alt="{\displaystyle t\in \mathbb {C} }" loading="lazy">, we have modified generating functions for these convolution polynomial sequences given by

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<annotation encoding="application/x-tex">{\displaystyle {\frac {zF_{n}(x+tn)}{(x+tn)}}=\left[z^{n}\right]{\mathcal {F}}_{t}(z)^{x},}</annotation>

</semantics>

</math>

where 𝓕t(z) is implicitly defined by a <a href="Functional_equation" title="Functional equation">functional equation</a> of the form 𝓕t(z) = F(x𝓕t(z)t). Moreover, we can use matrix methods (as in the reference) to prove that given two convolution polynomial sequences, ⟨ fn(x) ⟩ and ⟨ gn(x) ⟩, with respective corresponding generating functions, F(z)x and G(z)x, then for arbitrary t we have the identity

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<annotation encoding="application/x-tex">{\displaystyle \left[z^{n}\right]\left(G(z)F\left(zG(z)^{t}\right)\right)^{x}=\sum _{k=0}^{n}F_{k}(x)G_{n-k}(x+tk).}</annotation>

</semantics>

</math>

Examples of convolution polynomial sequences include the binomial power series, 𝓑t(z) = 1 + z𝓑t(z)t, so-termed tree polynomials, the <a href="Bell_numbers" class="mw-redirect" title="Bell numbers">Bell numbers</a>, B(n), the <a href="Laguerre_polynomials" title="Laguerre polynomials">Laguerre polynomials</a>, and the <a href="Stirling_polynomial" class="mw-redirect" title="Stirling polynomial">Stirling convolution polynomials</a>.

<div class="mw-heading mw-heading2"><h2 id="Ordinary_generating_functions">Ordinary generating functions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Examples_for_simple_sequences">Examples for simple sequences</h3></div>

Polynomials are a special case of ordinary generating functions, corresponding to finite sequences, or equivalently sequences that vanish after a certain point. These are important in that many finite sequences can usefully be interpreted as generating functions, such as the <a href="Poincar%C3%A9_polynomial" class="mw-redirect" title="Poincaré polynomial">Poincaré polynomial</a> and others.

A fundamental generating function is that of the constant sequence 1, 1, 1, 1, 1, 1, 1, 1, 1, ..., whose ordinary generating function is the <a href="Geometric_series#Closed-form_formula" title="Geometric series">geometric series</a>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }x^{n}={\frac {1}{1-x}}.}</annotation>

</semantics>

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The left-hand side is the <a href="Maclaurin_series" class="mw-redirect" title="Maclaurin series">Maclaurin series</a> expansion of the right-hand side. Alternatively, the equality can be justified by multiplying the power series on the left by 1 − x, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of x0 are equal to 0). Moreover, there can be no other power series with this property. The left-hand side therefore designates the <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> of 1 − x in the ring of power series.

Expressions for the ordinary generating function of other sequences are easily derived from this one. For instance, the substitution x → ax gives the generating function for the <a href="Geometric_progression" title="Geometric progression">geometric sequence</a> 1, a, a2, a3, ... for any constant a:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }(ax)^{n}={\frac {1}{1-ax}}.}</annotation>

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(The equality also follows directly from the fact that the left-hand side is the Maclaurin series expansion of the right-hand side.) In particular,

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}x^{n}={\frac {1}{1+x}}.}</annotation>

</semantics>

</math>

One can also introduce regular gaps in the sequence by replacing x by some power of x, so for instance for the sequence 1, 0, 1, 0, 1, 0, 1, 0, ... (which skips over x, x3, x5, ...) one gets the generating function

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }x^{2n}={\frac {1}{1-x^{2}}}.}</annotation>

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By squaring the initial generating function, or by finding the derivative of both sides with respect to x and making a change of running variable n → n + 1, one sees that the coefficients form the sequence 1, 2, 3, 4, 5, ..., so one has

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }(n+1)x^{n}={\frac {1}{(1-x)^{2}}},}</annotation>

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and the third power has as coefficients the <a href="Triangular_number" title="Triangular number">triangular numbers</a> 1, 3, 6, 10, 15, 21, ... whose term n is the <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a> (n + 2 2), so that

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\binom {n+2}{2}}x^{n}={\frac {1}{(1-x)^{3}}}.}</annotation>

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More generally, for any non-negative integer k and non-zero real value a, it is true that

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }a^{n}{\binom {n+k}{k}}x^{n}={\frac {1}{(1-ax)^{k+1}}}\,.}</annotation>

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Since

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<annotation encoding="application/x-tex">{\displaystyle 2{\binom {n+2}{2}}-3{\binom {n+1}{1}}+{\binom {n}{0}}=2{\frac {(n+1)(n+2)}{2}}-3(n+1)+1=n^{2},}</annotation>

</semantics>

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one can find the ordinary generating function for the sequence 0, 1, 4, 9, 16, ... of <a href="Square_number" title="Square number">square numbers</a> by linear combination of binomial-coefficient generating sequences:

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<annotation encoding="application/x-tex">{\displaystyle G(n^{2};x)=\sum _{n=0}^{\infty }n^{2}x^{n}={\frac {2}{(1-x)^{3}}}-{\frac {3}{(1-x)^{2}}}+{\frac {1}{1-x}}={\frac {x(x+1)}{(1-x)^{3}}}.}</annotation>

</semantics>

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We may also expand alternately to generate this same sequence of squares as a sum of derivatives of the <a href="Geometric_series" title="Geometric series">geometric series</a> in the following form:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G(n^{2};x)&=\sum _{n=0}^{\infty }n^{2}x^{n}\\[4px]&=\sum _{n=0}^{\infty }n(n-1)x^{n}+\sum _{n=0}^{\infty }nx^{n}\\[4px]&=x^{2}D^{2}\left[{\frac {1}{1-x}}\right]+xD\left[{\frac {1}{1-x}}\right]\\[4px]&={\frac {2x^{2}}{(1-x)^{3}}}+{\frac {x}{(1-x)^{2}}}={\frac {x(x+1)}{(1-x)^{3}}}.\end{aligned}}}</annotation>

</semantics>

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By induction, we can similarly show for positive integers m ≥ 1 that<a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a>

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<annotation encoding="application/x-tex">{\displaystyle n^{m}=\sum _{j=0}^{m}{\begin{Bmatrix}m\\j\end{Bmatrix}}{\frac {n!}{(n-j)!}},}</annotation>

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where {n k} denote the <a href="Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling numbers of the second kind</a> and where the generating function

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {n!}{(n-j)!}}\,z^{n}={\frac {j!\cdot z^{j}}{(1-z)^{j+1}}},}</annotation>

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so that we can form the analogous generating functions over the integral mth powers generalizing the result in the square case above. In particular, since we can write

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<annotation encoding="application/x-tex">{\displaystyle {\frac {z^{k}}{(1-z)^{k+1}}}=\sum _{i=0}^{k}{\binom {k}{i}}{\frac {(-1)^{k-i}}{(1-z)^{i+1}}},}</annotation>

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we can apply a well-known finite sum identity involving the <a href="Stirling_numbers" class="mw-redirect" title="Stirling numbers">Stirling numbers</a> to obtain that<a href="#cite_note-12">[12]</a>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }n^{m}z^{n}=\sum _{j=0}^{m}{\begin{Bmatrix}m+1\\j+1\end{Bmatrix}}{\frac {(-1)^{m-j}j!}{(1-z)^{j+1}}}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Rational_functions">Rational functions</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Linear_recursive_sequence" class="mw-redirect" title="Linear recursive sequence">Linear recursive sequence</a></div>

The ordinary generating function of a sequence can be expressed as a <a href="Rational_function" title="Rational function">rational function</a> (the ratio of two finite-degree polynomials) if and only if the sequence is a <a href="Linear_recursive_sequence" class="mw-redirect" title="Linear recursive sequence">linear recursive sequence</a> with constant coefficients; this generalizes the examples above. Conversely, every sequence generated by a fraction of polynomials satisfies a linear recurrence with constant coefficients; these coefficients are identical to the coefficients of the fraction denominator polynomial (so they can be directly read off). This observation shows it is easy to solve for generating functions of sequences defined by a linear <a href="Finite_difference_equation" class="mw-redirect" title="Finite difference equation">finite difference equation</a> with constant coefficients, and then hence, for explicit closed-form formulas for the coefficients of these generating functions. The prototypical example here is to derive <a href="Binet's_formula" class="mw-redirect" title="Binet's formula">Binet's formula</a> for the <a href="Fibonacci_numbers" class="mw-redirect" title="Fibonacci numbers">Fibonacci numbers</a> via generating function techniques.

We also notice that the class of rational generating functions precisely corresponds to the generating functions that enumerate quasi-polynomial sequences of the form <a href="#cite_note-GFLECT-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle f_{n}=p_{1}(n)\rho _{1}^{n}+\cdots +p_{\ell }(n)\rho _{\ell }^{n},}</annotation>

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where the reciprocal roots, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{i}\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle \rho _{i}\in \mathbb {C} }</annotation>

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</math><img src="./cb19b21a719205aa55c43020a9c2c68c396c1997.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.52ex; height:2.676ex;" alt="{\displaystyle \rho _{i}\in \mathbb {C} }" loading="lazy">, are fixed scalars and where pi(n) is a polynomial in n for all 1 ≤ i ≤ ℓ.

In general, <a href="Generating_function_transformation#Hadamard_products_and_diagonal_generating_functions" title="Generating function transformation">Hadamard products</a> of rational functions produce rational generating functions. Similarly, if

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<annotation encoding="application/x-tex">{\displaystyle F(s,t):=\sum _{m,n\geq 0}f(m,n)w^{m}z^{n}}</annotation>

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is a bivariate rational generating function, then its corresponding diagonal generating function,

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {diag} (F):=\sum _{n=0}^{\infty }f(n,n)z^{n},}</annotation>

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is algebraic. For example, if we let<a href="#cite_note-14">[14]</a>

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<annotation encoding="application/x-tex">{\displaystyle F(s,t):=\sum _{i,j\geq 0}{\binom {i+j}{i}}s^{i}t^{j}={\frac {1}{1-s-t}},}</annotation>

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then this generating function's diagonal coefficient generating function is given by the well-known OGF formula

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {diag} (F)=\sum _{n=0}^{\infty }{\binom {2n}{n}}z^{n}={\frac {1}{\sqrt {1-4z}}}.}</annotation>

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This result is computed in many ways, including <a href="Cauchy's_integral_formula" title="Cauchy's integral formula">Cauchy's integral formula</a> or <a href="Contour_integration" title="Contour integration">contour integration</a>, taking complex <a href="Residue_(complex_analysis)" title="Residue (complex analysis)">residues</a>, or by direct manipulations of <a href="Formal_power_series" title="Formal power series">formal power series</a> in two variables.

<div class="mw-heading mw-heading3"><h3 id="Operations_on_generating_functions">Operations on generating functions</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Multiplication_yields_convolution">Multiplication yields convolution</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Cauchy_product" title="Cauchy product">Cauchy product</a></div>

Multiplication of ordinary generating functions yields a discrete <a href="Convolution" title="Convolution">convolution</a> (the <a href="Cauchy_product" title="Cauchy product">Cauchy product</a>) of the sequences. For example, the sequence of cumulative sums (compare to the slightly more general <a href="Euler%E2%80%93Maclaurin_formula" title="Euler–Maclaurin formula">Euler–Maclaurin formula</a>)

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<annotation encoding="application/x-tex">{\displaystyle (a_{0},a_{0}+a_{1},a_{0}+a_{1}+a_{2},\ldots )}</annotation>

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of a sequence with ordinary generating function G(an; x) has the generating function

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<annotation encoding="application/x-tex">{\displaystyle G(a_{n};x)\cdot {\frac {1}{1-x}}}</annotation>

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because <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style>⁠1/1 − x⁠ is the ordinary generating function for the sequence (1, 1, ...). See also the <a href="#Convolution_(Cauchy_products)">section on convolutions</a> in the applications section of this article below for further examples of problem solving with convolutions of generating functions and interpretations.

<div class="mw-heading mw-heading4"><h4 id="Shifting_sequence_indices">Shifting sequence indices</h4></div>

For integers m ≥ 1, we have the following two analogous identities for the modified generating functions enumerating the shifted sequence variants of ⟨ gn − m ⟩ and ⟨ gn + m ⟩, respectively:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&z^{m}G(z)=\sum _{n=m}^{\infty }g_{n-m}z^{n}\\[4px]&{\frac {G(z)-g_{0}-g_{1}z-\cdots -g_{m-1}z^{m-1}}{z^{m}}}=\sum _{n=0}^{\infty }g_{n+m}z^{n}.\end{aligned}}}</annotation>

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<div class="mw-heading mw-heading4"><h4 id="Differentiation_and_integration_of_generating_functions">Differentiation and integration of generating functions</h4></div>

We have the following respective power series expansions for the first derivative of a generating function and its integral:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G'(z)&=\sum _{n=0}^{\infty }(n+1)g_{n+1}z^{n}\\[4px]z\cdot G'(z)&=\sum _{n=0}^{\infty }ng_{n}z^{n}\\[4px]\int _{0}^{z}G(t)\,dt&=\sum _{n=1}^{\infty }{\frac {g_{n-1}}{n}}z^{n}.\end{aligned}}}</annotation>

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The differentiation–multiplication operation of the second identity can be repeated k times to multiply the sequence by nk, but that requires alternating between differentiation and multiplication. If instead doing k differentiations in sequence, the effect is to multiply by the kth <a href="Falling_factorial" class="mw-redirect" title="Falling factorial">falling factorial</a>:

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<annotation encoding="application/x-tex">{\displaystyle z^{k}G^{(k)}(z)=\sum _{n=0}^{\infty }n^{\underline {k}}g_{n}z^{n}=\sum _{n=0}^{\infty }n(n-1)\dotsb (n-k+1)g_{n}z^{n}\quad {\text{for all }}k\in \mathbb {N} .}</annotation>

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Using the <a href="Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling numbers of the second kind</a>, that can be turned into another formula for multiplying by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle n^{k}}</annotation>

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</math><img src="./53af70c07e0932d85d2fdb70c56360544c3a0b5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.483ex; height:2.676ex;" alt="{\displaystyle n^{k}}" loading="lazy"> as follows (see the main article on <a href="Generating_function_transformation#Derivative_transformations" title="Generating function transformation">generating function transformations</a>):

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<annotation encoding="application/x-tex">{\displaystyle \sum _{j=0}^{k}{\begin{Bmatrix}k\\j\end{Bmatrix}}z^{j}F^{(j)}(z)=\sum _{n=0}^{\infty }n^{k}f_{n}z^{n}\quad {\text{for all }}k\in \mathbb {N} .}</annotation>

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A negative-order reversal of this sequence powers formula corresponding to the operation of repeated integration is defined by the <a href="Generating_function_transformation#Derivative_transformations" title="Generating function transformation">zeta series transformation</a> and its generalizations defined as a derivative-based <a href="Generating_function_transformation" title="Generating function transformation">transformation of generating functions</a>, or alternately termwise by and performing an <a href="Generating_function_transformation#Polylogarithm_series_transformations" title="Generating function transformation">integral transformation</a> on the sequence generating function. Related operations of performing <a href="Fractional_calculus" title="Fractional calculus">fractional integration</a> on a sequence generating function are discussed <a href="Generating_function_transformation#Fractional_integrals_and_derivatives" title="Generating function transformation">here</a>.

<div class="mw-heading mw-heading4"><h4 id="Enumerating_arithmetic_progressions_of_sequences">Enumerating arithmetic progressions of sequences</h4></div>

In this section we give formulas for generating functions enumerating the sequence {fan + b} given an ordinary generating function F(z), where a ≥ 2, 0 ≤ b < a, and a and b are integers (see the <a href="Generating_function_transformation" title="Generating function transformation">main article on transformations</a>). For a = 2, this is simply the familiar decomposition of a function into <a href="Even_and_odd_functions" title="Even and odd functions">even and odd parts</a> (i.e., even and odd powers):

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }f_{2n}z^{2n}&={\frac {F(z)+F(-z)}{2}}\\[4px]\sum _{n=0}^{\infty }f_{2n+1}z^{2n+1}&={\frac {F(z)-F(-z)}{2}}.\end{aligned}}}</annotation>

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More generally, suppose that a ≥ 3 and that ωa = exp ⁠2πi/a⁠ denotes the ath <a href="Root_of_unity" title="Root of unity">primitive root of unity</a>. Then, as an application of the <a href="Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a>, we have the formula<a href="#cite_note-TAOCPV1-15">[15]</a>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }f_{an+b}z^{an+b}={\frac {1}{a}}\sum _{m=0}^{a-1}\omega _{a}^{-mb}F\left(\omega _{a}^{m}z\right).}</annotation>

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For integers m ≥ 1, another useful formula providing somewhat reversed floored arithmetic progressions — effectively repeating each coefficient m times — are generated by the identity<a href="#cite_note-16">[16]</a>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }f_{\left\lfloor {\frac {n}{m}}\right\rfloor }z^{n}={\frac {1-z^{m}}{1-z}}F(z^{m})=\left(1+z+\cdots +z^{m-2}+z^{m-1}\right)F(z^{m}).}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="P-recursive_sequences_and_holonomic_generating_functions">P-recursive sequences and holonomic generating functions</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Definitions">Definitions</h4></div>

A formal power series (or function) F(z) is said to be holonomic if it satisfies a linear differential equation of the form<a href="#cite_note-17">[17]</a>

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<annotation encoding="application/x-tex">{\displaystyle c_{0}(z)F^{(r)}(z)+c_{1}(z)F^{(r-1)}(z)+\cdots +c_{r}(z)F(z)=0,}</annotation>

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where the coefficients ci(z) are in the field of rational functions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} (z)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} (z)}</annotation>

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</math><img src="./5917f17c3d7c66273521f830ef30a8950703188c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.576ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} (z)}" loading="lazy">. Equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(z)}">

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<annotation encoding="application/x-tex">{\displaystyle F(z)}</annotation>

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</math><img src="./5cf298d777120df944559c5e985b88a824debb80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.638ex; height:2.843ex;" alt="{\displaystyle F(z)}" loading="lazy"> is holonomic if the vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} (z)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} (z)}</annotation>

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Since we can clear denominators if need be in the previous equation, we may assume that the functions, ci(z) are polynomials in z. Thus we can see an equivalent condition that a generating function is holonomic if its coefficients satisfy a P-recurrence of the form

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {c}}_{s}(n)f_{n+s}+{\widehat {c}}_{s-1}(n)f_{n+s-1}+\cdots +{\widehat {c}}_{0}(n)f_{n}=0,}</annotation>

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for all large enough n ≥ n0 and where the ĉi(n) are fixed finite-degree polynomials in n. In other words, the properties that a sequence be P-recursive and have a holonomic generating function are equivalent. Holonomic functions are closed under the <a href="Generating_function_transformation#Hadamard_products_and_diagonal_generating_functions" title="Generating function transformation">Hadamard product</a> operation ⊙ on generating functions.

<div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4></div>

The functions ez, log z, cos z, arcsin z, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1+z}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {1+z}}}</annotation>

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</math><img src="./15eb2cb08e00208010c8524c7d374483b1f534dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.027ex; height:3.009ex;" alt="{\displaystyle {\sqrt {1+z}}}" loading="lazy">, the <a href="Dilogarithm" title="Dilogarithm">dilogarithm</a> function Li2(z), the <a href="Generalized_hypergeometric_function" title="Generalized hypergeometric function">generalized hypergeometric functions</a> pFq(...; ...; z) and the functions defined by the power series

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {z^{n}}{(n!)^{2}}}}</annotation>

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and the non-convergent

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }n!\cdot z^{n}}</annotation>

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are all holonomic.

Examples of P-recursive sequences with holonomic generating functions include fn ≔ ⁠1/n + 1⁠ (2n n) and fn ≔ ⁠2n/n2 + 1⁠, where sequences such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}}">

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</math><img src="./2a2994734eae382ce30100fb17b9447fd8e99f81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}}" loading="lazy"> and log n are not P-recursive due to the nature of singularities in their corresponding generating functions. Similarly, functions with infinitely many singularities such as tan z, sec z, and <a href="Gamma_function" title="Gamma function">Γ(z)</a> are not holonomic functions.

<div class="mw-heading mw-heading4"><h4 id="Software_for_working_with_P-recursive_sequences_and_holonomic_generating_functions">Software for working with P-recursive sequences and holonomic generating functions</h4></div>

Tools for processing and working with P-recursive sequences in <a href="Mathematica" class="mw-redirect" title="Mathematica">Mathematica</a> include the software packages provided for non-commercial use on the <a rel="nofollow" class="external text" href="https://www.risc.jku.at/research/combinat/software/">RISC Combinatorics Group algorithmic combinatorics software</a> site. Despite being mostly closed-source, particularly powerful tools in this software suite are provided by the <code>Guess</code> package for guessing P-recurrences for arbitrary input sequences (useful for <a href="Experimental_mathematics" title="Experimental mathematics">experimental mathematics</a> and exploration) and the <code>Sigma</code> package which is able to find P-recurrences for many sums and solve for closed-form solutions to P-recurrences involving generalized <a href="Harmonic_number" title="Harmonic number">harmonic numbers</a>.<a href="#cite_note-18">[18]</a> Other packages listed on this particular RISC site are targeted at working with holonomic generating functions specifically.

<div class="mw-heading mw-heading3"><h3 id="Relation_to_discrete-time_Fourier_transform">Relation to discrete-time Fourier transform</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a></div>

When the series <a href="Absolute_convergence" title="Absolute convergence">converges absolutely</a>,

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<annotation encoding="application/x-tex">{\displaystyle G\left(a_{n};e^{-i\omega }\right)=\sum _{n=0}^{\infty }a_{n}e^{-i\omega n}}</annotation>

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is the discrete-time Fourier transform of the sequence a0, a1, ....

<div class="mw-heading mw-heading3"><h3 id="Asymptotic_growth_of_a_sequence">Asymptotic growth of a sequence</h3></div>

In calculus, often the growth rate of the coefficients of a power series can be used to deduce a <a href="Radius_of_convergence" title="Radius of convergence">radius of convergence</a> for the power series. The reverse can also hold; often the radius of convergence for a generating function can be used to deduce the <a href="Asymptotic_analysis" title="Asymptotic analysis">asymptotic growth</a> of the underlying sequence.

For instance, if an ordinary generating function G(an; x) that has a finite radius of convergence of r can be written as

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<annotation encoding="application/x-tex">{\displaystyle G(a_{n};x)={\frac {A(x)+B(x)\left(1-{\frac {x}{r}}\right)^{-\beta }}{x^{\alpha }}}}</annotation>

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where each of A(x) and B(x) is a function that is <a href="Analytic_function" title="Analytic function">analytic</a> to a radius of convergence greater than r (or is <a href="Entire_function" title="Entire function">entire</a>), and where B(r) ≠ 0 then

<msub>

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</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

<mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}\sim {\frac {B(r)}{r^{\alpha }}}{\binom {n+\beta -1}{n}}\left({\frac {1}{r}}\right)^{n}={\frac {B(r)}{r^{\alpha }}}\left(\!\!{\binom {\beta }{n}}\!\!\right)\left({\frac {1}{r}}\right)^{n}\,,}</annotation>

</semantics>

</math>

using the <a href="Gamma_function" title="Gamma function">gamma function</a>, a <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficient</a>, or a <a href="Multiset_coefficient" class="mw-redirect" title="Multiset coefficient">multiset coefficient</a>. Note that limit as n goes to infinity of the ratio of an to any of these expressions is guaranteed to be 1; not merely that an is proportional to them.

Often this approach can be iterated to generate several terms in an asymptotic series for an. In particular,

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\left(a_{n}-{\frac {B(r)}{r^{\alpha }}}{\binom {n+\beta -1}{n}}\left({\frac {1}{r}}\right)^{n};x\right)=G(a_{n};x)-{\frac {B(r)}{r^{\alpha }}}\left(1-{\frac {x}{r}}\right)^{-\beta }\,.}</annotation>

</semantics>

</math>

The asymptotic growth of the coefficients of this generating function can then be sought via the finding of A, B, α, β, and r to describe the generating function, as above.

Similar asymptotic analysis is possible for exponential generating functions; with an exponential generating function, it is ⁠an/n!⁠ that grows according to these asymptotic formulae. Generally, if the generating function of one sequence minus the generating function of a second sequence has a radius of convergence that is larger than the radius of convergence of the individual generating functions then the two sequences have the same asymptotic growth.

<div class="mw-heading mw-heading4"><h4 id="Asymptotic_growth_of_the_sequence_of_squares">Asymptotic growth of the sequence of squares</h4></div>

As derived above, the ordinary generating function for the sequence of squares is:

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(n^{2};x)={\frac {x(x+1)}{(1-x)^{3}}}.}</annotation>

</semantics>

</math>

With r = 1, α = −1, β = 3, A(x) = 0, and B(x) = x + 1, we can verify that the squares grow as expected, like the squares:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

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<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}={\frac {1+1}{1^{-1}\,\Gamma (3)}}\,n^{3-1}\left({\frac {1}{1}}\right)^{n}=n^{2}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Asymptotic_growth_of_the_Catalan_numbers">Asymptotic growth of the Catalan numbers</h4></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Catalan_number" title="Catalan number">Catalan number</a></div>

The ordinary generating function for the <a href="Catalan_number" title="Catalan number">Catalan numbers</a> is

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

<mrow>

</mrow>

</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(C_{n};x)={\frac {1-{\sqrt {1-4x}}}{2x}}.}</annotation>

</semantics>

</math>

With r = ⁠1/4⁠, α = 1, β = −⁠1/2⁠, A(x) = ⁠1/2⁠, and B(x) = −⁠1/2⁠, we can conclude that, for the Catalan numbers:

<msub>

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</msup>

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</msup>

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<mfrac>

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<mrow>

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</msup>

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<msup>

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</mrow>

</msup>

<msup>

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<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

<mfrac>

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</msup>

<msqrt>

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</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{n}\sim {\frac {B(r)}{r^{\alpha }\Gamma (\beta )}}\,n^{\beta -1}\left({\frac {1}{r}}\right)^{n}={\frac {-{\frac {1}{2}}}{\left({\frac {1}{4}}\right)^{1}\Gamma \left(-{\frac {1}{2}}\right)}}\,n^{-{\frac {1}{2}}-1}\left({\frac {1}{\,{\frac {1}{4}}\,}}\right)^{n}={\frac {4^{n}}{n^{\frac {3}{2}}{\sqrt {\pi }}}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Bivariate_and_multivariate_generating_functions">Bivariate and multivariate generating functions</h3></div>

The generating function in several variables can be generalized to arrays with multiple indices. These non-polynomial double sum examples are called multivariate generating functions, or super generating functions. For two variables, these are often called bivariate generating functions.

<div class="mw-heading mw-heading4"><h4 id="Bivariate_case">Bivariate case</h4></div>

The ordinary generating function of a two-dimensional array am,n (where n and m are natural numbers) is:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(a_{m,n};x,y)=\sum _{m,n=0}^{\infty }a_{m,n}x^{m}y^{n}.}</annotation>

</semantics>

</math>For instance, since (1 + x)n is the ordinary generating function for <a href="Binomial_coefficients" class="mw-redirect" title="Binomial coefficients">binomial coefficients</a> for a fixed n, one may ask for a bivariate generating function that generates the binomial coefficients (n k) for all k and n. To do this, consider (1 + x)n itself as a sequence in n, and find the generating function in y that has these sequence values as coefficients. Since the generating function for an is:

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-ay}},}</annotation>

</semantics>

</math>the generating function for the binomial coefficients is:

</mrow>

</munder>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

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</mfrac>

</mrow>

<mfrac>

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</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n,k}{\binom {n}{k}}x^{k}y^{n}={\frac {1}{1-(1+x)y}}={\frac {1}{1-y-xy}}.}</annotation>

</semantics>

</math>Other examples of such include the following two-variable generating functions for the <a href="Binomial_coefficients" class="mw-redirect" title="Binomial coefficients">binomial coefficients</a>, the <a href="Stirling_numbers" class="mw-redirect" title="Stirling numbers">Stirling numbers</a>, and the <a href="Eulerian_numbers" class="mw-redirect" title="Eulerian numbers">Eulerian numbers</a>, where ω and z denote the two variables:<a href="#cite_note-19">[19]</a>

<mtr>

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</munder>

<mrow>

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<mtr>

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<msup>

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</msup>

<mfrac>

<msup>

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</msup>

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</mfrac>

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<mfrac>

<mrow>

<msup>

</mrow>

</msup>

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</mfrac>

</mrow>

</mtd>

<mtd>

</mrow>

</munder>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

</mrow>

</munder>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

<mtd>

</mrow>

</munder>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e^{z+wz}&=\sum _{m,n\geq 0}{\binom {n}{m}}w^{m}{\frac {z^{n}}{n!}}\\[4px]e^{w(e^{z}-1)}&=\sum _{m,n\geq 0}{\begin{Bmatrix}n\\m\end{Bmatrix}}w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {1}{(1-z)^{w}}}&=\sum _{m,n\geq 0}{\begin{bmatrix}n\\m\end{bmatrix}}w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {1-w}{e^{(w-1)z}-w}}&=\sum _{m,n\geq 0}\left\langle {\begin{matrix}n\\m\end{matrix}}\right\rangle w^{m}{\frac {z^{n}}{n!}}\\[4px]{\frac {e^{w}-e^{z}}{we^{z}-ze^{w}}}&=\sum _{m,n\geq 0}\left\langle {\begin{matrix}m+n+1\\m\end{matrix}}\right\rangle {\frac {w^{m}z^{n}}{(m+n+1)!}}.\end{aligned}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Multivariate_case">Multivariate case</h4></div>

Multivariate generating functions arise in practice when calculating the number of <a href="Contingency_tables" class="mw-redirect" title="Contingency tables">contingency tables</a> of non-negative integers with specified row and column totals. Suppose the table has r rows and c columns; the row sums are t1, t2 ... tr and the column sums are s1, s2 ... sc. Then, according to <a href="I._J._Good" title="I. J. Good">I. J. Good</a>,<a href="#cite_note-Good_1986-20">[20]</a> the number of such tables is the coefficient of:

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

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</mrow>

<msub>

</mrow>

</msub>

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</mrow>

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{1}^{t_{1}}\cdots x_{r}^{t_{r}}y_{1}^{s_{1}}\cdots y_{c}^{s_{c}}}</annotation>

</semantics>

</math>in:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{i=1}^{r}\prod _{j=1}^{c}{\frac {1}{1-x_{i}y_{j}}}.}">

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{i=1}^{r}\prod _{j=1}^{c}{\frac {1}{1-x_{i}y_{j}}}.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading3"><h3 id="Representation_by_continued_fractions_(Jacobi-type_J-fractions)">Representation by continued fractions (Jacobi-type J-fractions)</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Definitions_2">Definitions</h4></div>

Expansions of (formal) Jacobi-type and Stieltjes-type <a href="Generalized_continued_fraction" class="mw-redirect" title="Generalized continued fraction">continued fractions</a> (J-fractions and S-fractions, respectively) whose hth rational convergents represent <a href="Order_of_accuracy" title="Order of accuracy">2h-order accurate</a> power series are another way to express the typically divergent ordinary generating functions for many special one and two-variate sequences. The particular form of the Jacobi-type continued fractions (J-fractions) are expanded as in the following equation and have the next corresponding power series expansions with respect to z for some specific, application-dependent component sequences, {abi} and {ci}, where z ≠ 0 denotes the formal variable in the second power series expansion given below:<a href="#cite_note-21">[21]</a>

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</mpadded>

<msub>

</mrow>

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<mfrac>

<mrow>

</mpadded>

<msub>

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</mrow>

</msub>

<msup>

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</msup>

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</mstyle>

</mrow>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

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</mrow>

</mrow>

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</msubsup>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

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<msub>

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</msub>

</mrow>

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<msup>

</mrow>

</msup>

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</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J^{[\infty ]}(z)&={\cfrac {1}{1-c_{1}z-{\cfrac {{\text{ab}}_{2}z^{2}}{1-c_{2}z-{\cfrac {{\text{ab}}_{3}z^{2}}{\ddots }}}}}}\\[4px]&=1+c_{1}z+\left({\text{ab}}_{2}+c_{1}^{2}\right)z^{2}+\left(2{\text{ab}}_{2}c_{1}+c_{1}^{3}+{\text{ab}}_{2}c_{2}\right)z^{3}+\cdots \end{aligned}}}</annotation>

</semantics>

</math>

The coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{n}}</annotation>

</semantics>

</math><img src="./5b1a8cdd7ee39054e510deeb38ee551cc7616ae1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.309ex; height:2.343ex;" alt="{\displaystyle z^{n}}" loading="lazy">, denoted in shorthand by jn ≔ [zn] J[∞](z), in the previous equations correspond to matrix solutions of the equations:

<mrow>

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<mtr>

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</msub>

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<mtd>

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<mtd>

<msub>

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<mtd>

</mtd>

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<mtr>

<mtd>

</mtd>

<mtd>

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<mtd>

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<mtd>

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</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

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</mtr>

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<mtd>

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</mtr>

</mtable>

</mrow>

<mrow>

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<mtd>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}k_{0,1}&k_{1,1}&0&0&\cdots \\k_{0,2}&k_{1,2}&k_{2,2}&0&\cdots \\k_{0,3}&k_{1,3}&k_{2,3}&k_{3,3}&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}}={\begin{bmatrix}k_{0,0}&0&0&0&\cdots \\k_{0,1}&k_{1,1}&0&0&\cdots \\k_{0,2}&k_{1,2}&k_{2,2}&0&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}}\cdot {\begin{bmatrix}c_{1}&1&0&0&\cdots \\{\text{ab}}_{2}&c_{2}&1&0&\cdots \\0&{\text{ab}}_{3}&c_{3}&1&\cdots \\\vdots &\vdots &\vdots &\vdots \end{bmatrix}},}</annotation>

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where j0 ≡ k0,0 = 1, jn = k0,n for n ≥ 1, kr,s = 0 if r > s, and where for all integers p, q ≥ 0, we have an addition formula relation given by:

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<annotation encoding="application/x-tex">{\displaystyle j_{p+q}=k_{0,p}\cdot k_{0,q}+\sum _{i=1}^{\min(p,q)}{\text{ab}}_{2}\cdots {\text{ab}}_{i+1}\times k_{i,p}\cdot k_{i,q}.}</annotation>

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</math>

<div class="mw-heading mw-heading4"><h4 id="Properties_of_the_hth_convergent_functions">Properties of the hth convergent functions</h4></div>

For h ≥ 0 (though in practice when h ≥ 2), we can define the rational hth convergents to the infinite J-fraction, J[∞](z), expanded by:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Conv} _{h}(z):={\frac {P_{h}(z)}{Q_{h}(z)}}=j_{0}+j_{1}z+\cdots +j_{2h-1}z^{2h-1}+\sum _{n=2h}^{\infty }{\widetilde {j}}_{h,n}z^{n}}</annotation>

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component-wise through the sequences, Ph(z) and Qh(z), defined recursively by:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P_{h}(z)&=(1-c_{h}z)P_{h-1}(z)-{\text{ab}}_{h}z^{2}P_{h-2}(z)+\delta _{h,1}\\Q_{h}(z)&=(1-c_{h}z)Q_{h-1}(z)-{\text{ab}}_{h}z^{2}Q_{h-2}(z)+(1-c_{1}z)\delta _{h,1}+\delta _{0,1}.\end{aligned}}}</annotation>

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</math>

Moreover, the rationality of the convergent function Convh(z) for all h ≥ 2 implies additional finite difference equations and congruence properties satisfied by the sequence of jn, and for Mh ≔ ab2 ⋯ abh + 1 if h ‖ Mh then we have the congruence

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<annotation encoding="application/x-tex">{\displaystyle j_{n}\equiv [z^{n}]\operatorname {Conv} _{h}(z){\pmod {h}},}</annotation>

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for non-symbolic, determinate choices of the parameter sequences {abi} and {ci} when h ≥ 2, that is, when these sequences do not implicitly depend on an auxiliary parameter such as q, x, or R as in the examples contained in the table below.

<div class="mw-heading mw-heading4"><h4 id="Examples_2">Examples</h4></div>

The next table provides examples of closed-form formulas for the component sequences found computationally (and subsequently proved correct in the cited references<a href="#cite_note-22">[22]</a>)

in several special cases of the prescribed sequences, jn, generated by the general expansions of the J-fractions defined in the first subsection. Here we define 0 < |a|, |b|, |q| < 1 and the parameters <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,\alpha \in \mathbb {Z} ^{+}}">

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<annotation encoding="application/x-tex">{\displaystyle R,\alpha \in \mathbb {Z} ^{+}}</annotation>

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</math><img src="./0dcd7d07bc0704a132024cbb675ce79aa7d292f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.187ex; height:2.843ex;" alt="{\displaystyle R,\alpha \in \mathbb {Z} ^{+}}" loading="lazy"> and x to be indeterminates with respect to these expansions, where the prescribed sequences enumerated by the expansions of these J-fractions are defined in terms of the <a href="Q-Pochhammer_symbol" title="Q-Pochhammer symbol">q-Pochhammer symbol</a>, <a href="Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a>, and the <a href="Binomial_coefficients" class="mw-redirect" title="Binomial coefficients">binomial coefficients</a>.

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<annotation encoding="application/x-tex">{\displaystyle j_{n}}</annotation>

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</math><img src="./59cafe09467c3b2a6aa18cbf0218f2c8f2af233e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.203ex; height:2.509ex;" alt="{\displaystyle j_{n}}" loading="lazy"></th>

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<annotation encoding="application/x-tex">{\displaystyle c_{1}}</annotation>

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</math><img src="./77b7dc6d279091d354e0b90889b463bfa7eb7247.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{1}}" loading="lazy"></th>

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<annotation encoding="application/x-tex">{\displaystyle c_{i}(i\geq 2)}</annotation>

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</math><img src="./ce515bcdb92513a1fb01a473ac42f56bf866fa9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.679ex; height:2.843ex;" alt="{\displaystyle c_{i}(i\geq 2)}" loading="lazy"></th>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {ab} _{i}(i\geq 2)}</annotation>

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</math><img src="./7b00de46c913d194d0bd49e51b6cc8d218b76d79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.127ex; height:2.843ex;" alt="{\displaystyle \mathrm {ab} _{i}(i\geq 2)}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle q^{n^{2}}}</annotation>

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</math><img src="./a94913eda8b7f2758179098c83a5e58055b8deaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.129ex; height:3.343ex;" alt="{\displaystyle q^{n^{2}}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle q^{2h-3}\left(q^{2h}+q^{2h-2}-1\right)}</annotation>

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</math><img src="./b61e79b425446d3ea1b6e1e9db4b3a738c22d371.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.803ex; height:3.343ex;" alt="{\displaystyle q^{2h-3}\left(q^{2h}+q^{2h-2}-1\right)}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle q^{6h-10}\left(q^{2h-2}-1\right)}</annotation>

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</math><img src="./86e9ba3a296b05a3bfff95437d92a1fceee3e32a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.704ex; height:3.343ex;" alt="{\displaystyle q^{6h-10}\left(q^{2h-2}-1\right)}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle (a;q)_{n}}</annotation>

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</math><img src="./3a132fadd647f6ffcf79efb2ec0534e64aaca3e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.361ex; height:2.843ex;" alt="{\displaystyle (a;q)_{n}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle 1-a}</annotation>

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</math><img src="./c342073d5dde33970fc5ae7c6f06a908a991a6ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.233ex; height:2.343ex;" alt="{\displaystyle 1-a}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle q^{h-1}-aq^{h-2}\left(q^{h}+q^{h-1}-1\right)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle aq^{2h-4}\left(aq^{h-2}-1\right)\left(q^{h-1}-1\right)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \left(zq^{-n};q\right)_{n}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {q-z}{q}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {q^{h}-z-qz+q^{h}z}{q^{2h-1}}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\left(q^{h-1}-1\right)\left(q^{h-1}-z\right)\cdot z}{q^{4h-5}}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {(a;q)_{n}}{(b;q)_{n}}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1-a}{1-b}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {q^{i-2}\left(q+abq^{2i-3}+a(1-q^{i-1}-q^{i})+b(q^{i}-q-1)\right)}{\left(1-bq^{2i-4}\right)\left(1-bq^{2i-2}\right)}}}</annotation>

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</math><img src="./98184553eaf23e32a03eb6093fbac6c798e77404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.292ex; height:6.843ex;" alt="{\displaystyle {\frac {q^{i-2}\left(q+abq^{2i-3}+a(1-q^{i-1}-q^{i})+b(q^{i}-q-1)\right)}{\left(1-bq^{2i-4}\right)\left(1-bq^{2i-2}\right)}}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {q^{2i-4}\left(1-bq^{i-3}\right)\left(1-aq^{i-2}\right)\left(a-bq^{i-2}\right)\left(1-q^{i-1}\right)}{\left(1-bq^{2i-5}\right)\left(1-bq^{2i-4}\right)^{2}\left(1-bq^{2i-3}\right)}}}</annotation>

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</math><img src="./409ba47f4c63bba7d277766676139e1de3c3f73d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.928ex; height:7.343ex;" alt="{\displaystyle {\frac {q^{2i-4}\left(1-bq^{i-3}\right)\left(1-aq^{i-2}\right)\left(a-bq^{i-2}\right)\left(1-q^{i-1}\right)}{\left(1-bq^{2i-5}\right)\left(1-bq^{2i-4}\right)^{2}\left(1-bq^{2i-3}\right)}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \alpha ^{n}\cdot \left({\frac {R}{\alpha }}\right)_{n}}</annotation>

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</math><img src="./4bf56d57113459e78d34f68bfaf96abdd51f2176.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.625ex; height:6.176ex;" alt="{\displaystyle \alpha ^{n}\cdot \left({\frac {R}{\alpha }}\right)_{n}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R+2\alpha (i-1)}</annotation>

</semantics>

</math><img src="./f639e71f1277799a5d310a322e4a317f7c1f2220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.869ex; height:2.843ex;" alt="{\displaystyle R+2\alpha (i-1)}" loading="lazy"></td>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (i-1)\alpha {\bigl (}R+(i-2)\alpha {\bigr )}}</annotation>

</semantics>

</math><img src="./dfa3d1e134ae5c3b9cd832e5f641d764cf118d1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.939ex; height:3.176ex;" alt="{\displaystyle (i-1)\alpha {\bigl (}R+(i-2)\alpha {\bigr )}}" loading="lazy">

</td></tr>

<tr>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-1)^{n}{\binom {x}{n}}}</annotation>

</semantics>

</math><img src="./329ef6d4ad4cfe4989264a7fc6cb446a2a46c901.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.814ex; height:6.176ex;" alt="{\displaystyle (-1)^{n}{\binom {x}{n}}}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -x}</annotation>

</semantics>

</math><img src="./ae55e66aeffc525917eed885b4b753ba5a7f8b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle -x}" loading="lazy"></td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -{\frac {(x+2(i-1)^{2})}{(2i-1)(2i-3)}}}</annotation>

</semantics>

</math><img src="./de0c192b09031df6047478c06054dd82ead898bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.198ex; height:6.676ex;" alt="{\displaystyle -{\frac {(x+2(i-1)^{2})}{(2i-1)(2i-3)}}}" loading="lazy">

</td>

<mrow>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}-{\dfrac {(x-i+2)(x+i-1)}{4\cdot (2i-3)^{2}}}&{\text{for }}i\geq 3;\\[4px]-{\frac {1}{2}}x(x+1)&{\text{for }}i=2.\end{cases}}}</annotation>

</semantics>

</math><img src="./d2863efa002e10d228d6012c3c7dcf9d9e5c03f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:38.43ex; height:10.843ex;" alt="{\displaystyle {\begin{cases}-{\dfrac {(x-i+2)(x+i-1)}{4\cdot (2i-3)^{2}}}&{\text{for }}i\geq 3;\\[4px]-{\frac {1}{2}}x(x+1)&{\text{for }}i=2.\end{cases}}}" loading="lazy">

</td></tr>

<tr>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-1)^{n}{\binom {x+n}{n}}}</annotation>

</semantics>

</math><img src="./a415011f9f1b166e83e8673a4314e21836bf89b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.984ex; height:6.176ex;" alt="{\displaystyle (-1)^{n}{\binom {x+n}{n}}}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -(x+1)}</annotation>

</semantics>

</math><img src="./b1e2106075c786e8568003a498bad678028bdead.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.95ex; height:2.843ex;" alt="{\displaystyle -(x+1)}" loading="lazy"></td>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {{\bigl (}x-2i(i-2)-1{\bigr )}}{(2i-1)(2i-3)}}}</annotation>

</semantics>

</math><img src="./c2afd6cf2f612d4fb739355e314fedf1780344c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.718ex; height:6.843ex;" alt="{\displaystyle {\frac {{\bigl (}x-2i(i-2)-1{\bigr )}}{(2i-1)(2i-3)}}}" loading="lazy">

</td>

<mrow>

<mtr>

<mtd>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

</semantics>

</td></tr></tbody></table>

The radii of convergence of these series corresponding to the definition of the Jacobi-type J-fractions given above are in general different from that of the corresponding power series expansions defining the ordinary generating functions of these sequences.

<div class="mw-heading mw-heading2"><h2 id="Examples_3">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Square_numbers">Square numbers</h3></div>

Generating functions for the sequence of <a href="Square_number" title="Square number">square numbers</a> an = n2 are:

<th>Generating function type</th>

<th>Equation

</th></tr>

<tr>

<td>Ordinary generating function</td>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(n^{2};x)=\sum _{n=0}^{\infty }n^{2}x^{n}={\frac {x(x+1)}{(1-x)^{3}}}}</annotation>

</semantics>

</math><img src="./c69b1bcee224e3c6009e9192f09c47b431f8bf61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.692ex; height:6.843ex;" alt="{\displaystyle G(n^{2};x)=\sum _{n=0}^{\infty }n^{2}x^{n}={\frac {x(x+1)}{(1-x)^{3}}}}" loading="lazy">

</td></tr>

<tr>

<td>Exponential generating function</td>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {EG} (n^{2};x)=\sum _{n=0}^{\infty }{\frac {n^{2}x^{n}}{n!}}=x(x+1)e^{x}}</annotation>

</semantics>

</math><img src="./35cc6f6056f7ee3631e0edb68f3fcc3dd4237fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.529ex; height:6.843ex;" alt="{\displaystyle \operatorname {EG} (n^{2};x)=\sum _{n=0}^{\infty }{\frac {n^{2}x^{n}}{n!}}=x(x+1)e^{x}}" loading="lazy">

</td></tr>

<tr>

<td>Bell series</td>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {BG} _{p}\left(n^{2};x\right)=\sum _{n=0}^{\infty }\left(p^{n}\right)^{2}x^{n}={\frac {1}{1-p^{2}x}}}</annotation>

</semantics>

</math><img src="./4b47657d57c3595f32e31cedf231b9c9e364fbbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.602ex; height:6.843ex;" alt="{\displaystyle \operatorname {BG} _{p}\left(n^{2};x\right)=\sum _{n=0}^{\infty }\left(p^{n}\right)^{2}x^{n}={\frac {1}{1-p^{2}x}}}" loading="lazy">

</td></tr>

<tr>

<td>Dirichlet series</td>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {DG} \left(n^{2};s\right)=\sum _{n=1}^{\infty }{\frac {n^{2}}{n^{s}}}=\zeta (s-2)}</annotation>

</semantics>

</math><img src="./34277b6b18a85810a8f3c87cb6f56954ae9a9fed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.525ex; height:6.843ex;" alt="{\displaystyle \operatorname {DG} \left(n^{2};s\right)=\sum _{n=1}^{\infty }{\frac {n^{2}}{n^{s}}}=\zeta (s-2)}" loading="lazy">

</td></tr></tbody></table>

where ζ(s) is the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Generating functions are used to:

<ul><li>Find a <a href="Closed_formula" class="mw-redirect" title="Closed formula">closed formula</a> for a sequence given in a recurrence relation, for example, <a href="Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a>.</li>

<li>Find <a href="Recurrence_relation" title="Recurrence relation">recurrence relations</a> for sequences—the form of a generating function may suggest a recurrence formula.</li>

<li>Find relationships between sequences—if the generating functions of two sequences have a similar form, then the sequences themselves may be related.</li>

<li>Explore the asymptotic behaviour of sequences.</li>

<li>Prove identities involving sequences.</li>

<li>Solve <a href="Enumeration" title="Enumeration">enumeration</a> problems in <a href="Combinatorics" title="Combinatorics">combinatorics</a> and encoding their solutions. <a href="Rook_polynomial" title="Rook polynomial">Rook polynomials</a> are an example of an application in combinatorics.</li>

<li>Evaluate infinite sums.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Various_techniques:_Evaluating_sums_and_tackling_other_problems_with_generating_functions">Various techniques: Evaluating sums and tackling other problems with generating functions</h3></div>

<div class="mw-heading mw-heading4"><h4 id="Example_1:_Formula_for_sums_of_harmonic_numbers">Example 1: Formula for sums of harmonic numbers</h4></div>

Generating functions give us several methods to manipulate sums and to establish identities between sums.

The simplest case occurs when sn = Σn k = 0 ak. We then know that S(z) = ⁠A(z)/1 − z⁠ for the corresponding ordinary generating functions.

For example, we can manipulate

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{n}=\sum _{k=1}^{n}H_{k}\,,}</annotation>

</semantics>

</math>

where Hk = 1 + ⁠1/2⁠ + ⋯ + ⁠1/k⁠ are the <a href="Harmonic_number" title="Harmonic number">harmonic numbers</a>. Let

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(z)=\sum _{n=1}^{\infty }{H_{n}z^{n}}}</annotation>

</semantics>

</math>

be the ordinary generating function of the harmonic numbers. Then

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(z)={\frac {1}{1-z}}\sum _{n=1}^{\infty }{\frac {z^{n}}{n}}\,,}</annotation>

</semantics>

</math>

and thus

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(z)=\sum _{n=1}^{\infty }{s_{n}z^{n}}={\frac {1}{(1-z)^{2}}}\sum _{n=1}^{\infty }{\frac {z^{n}}{n}}\,.}</annotation>

</semantics>

</math>

Using

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(1-z)^{2}}}=\sum _{n=0}^{\infty }(n+1)z^{n}\,,}</annotation>

</semantics>

</math>

<a href="#Convolution_(Cauchy_products)">convolution</a> with the numerator yields

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{n}=\sum _{k=1}^{n}{\frac {n+1-k}{k}}=(n+1)H_{n}-n\,,}</annotation>

</semantics>

</math>

which can also be written as

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{k=1}^{n}{H_{k}}=(n+1)(H_{n+1}-1)\,.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Example_2:_Modified_binomial_coefficient_sums_and_the_binomial_transform">Example 2: Modified binomial coefficient sums and the binomial transform</h4></div>

As another example of using generating functions to relate sequences and manipulate sums, for an arbitrary sequence ⟨ fn ⟩ we define the two sequences of sums

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{n}&:=\sum _{m=0}^{n}{\binom {n}{m}}f_{m}3^{n-m}\\[4px]{\tilde {s}}_{n}&:=\sum _{m=0}^{n}{\binom {n}{m}}(m+1)(m+2)(m+3)f_{m}3^{n-m}\,,\end{aligned}}}</annotation>

</semantics>

</math>

for all n ≥ 0, and seek to express the second sums in terms of the first. We suggest an approach by generating functions.

First, we use the <a href="Binomial_transform" title="Binomial transform">binomial transform</a> to write the generating function for the first sum as

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(z)={\frac {1}{1-3z}}F\left({\frac {z}{1-3z}}\right).}</annotation>

</semantics>

</math>

Since the generating function for the sequence ⟨ (n + 1)(n + 2)(n + 3) fn ⟩ is given by

<msup>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 6F(z)+18zF'(z)+9z^{2}F''(z)+z^{3}F'''(z)}</annotation>

</semantics>

</math>

we may write the generating function for the second sum defined above in the form

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<msup>

</msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {S}}(z)={\frac {6}{(1-3z)}}F\left({\frac {z}{1-3z}}\right)+{\frac {18z}{(1-3z)^{2}}}F'\left({\frac {z}{1-3z}}\right)+{\frac {9z^{2}}{(1-3z)^{3}}}F''\left({\frac {z}{1-3z}}\right)+{\frac {z^{3}}{(1-3z)^{4}}}F'''\left({\frac {z}{1-3z}}\right).}</annotation>

</semantics>

</math>

In particular, we may write this modified sum generating function in the form of

<msup>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a(z)\cdot S(z)+b(z)\cdot zS'(z)+c(z)\cdot z^{2}S''(z)+d(z)\cdot z^{3}S'''(z),}</annotation>

</semantics>

</math>

for a(z) = 6(1 − 3z)3, b(z) = 18(1 − 3z)3, c(z) = 9(1 − 3z)3, and d(z) = (1 − 3z)3, where (1 − 3z)3 = 1 − 9z + 27z2 − 27z3.

Finally, it follows that we may express the second sums through the first sums in the following form:

<mtr>

<mtd>

<msub>

<mover>

</mover>

</mrow>

</mrow>

</msub>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\tilde {s}}_{n}&=[z^{n}]\left(6(1-3z)^{3}\sum _{n=0}^{\infty }s_{n}z^{n}+18(1-3z)^{3}\sum _{n=0}^{\infty }ns_{n}z^{n}+9(1-3z)^{3}\sum _{n=0}^{\infty }n(n-1)s_{n}z^{n}+(1-3z)^{3}\sum _{n=0}^{\infty }n(n-1)(n-2)s_{n}z^{n}\right)\\[4px]&=(n+1)(n+2)(n+3)s_{n}-9n(n+1)(n+2)s_{n-1}+27(n-1)n(n+1)s_{n-2}-(n-2)(n-1)ns_{n-3}.\end{aligned}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Example_3:_Generating_functions_for_mutually_recursive_sequences">Example 3: Generating functions for mutually recursive sequences</h4></div>

In this example, we reformulate a generating function example given in Section 7.3 of Concrete Mathematics (see also Section 7.1 of the same reference for pretty pictures of generating function series). In particular, suppose that we seek the total number of ways (denoted Un) to tile a 3-by-n rectangle with unmarked 2-by-1 domino pieces. Let the auxiliary sequence, Vn, be defined as the number of ways to cover a 3-by-n rectangle-minus-corner section of the full rectangle. We seek to use these definitions to give a <a href="Closed-form_expression" title="Closed-form expression">closed form</a> formula for Un without breaking down this definition further to handle the cases of vertical versus horizontal dominoes. Notice that the ordinary generating functions for our two sequences correspond to the series:

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U(z)=1+3z^{2}+11z^{4}+41z^{6}+\cdots ,\\V(z)=z+4z^{3}+15z^{5}+56z^{7}+\cdots .\end{aligned}}}</annotation>

</semantics>

</math>

If we consider the possible configurations that can be given starting from the left edge of the 3-by-n rectangle, we are able to express the following mutually dependent, or mutually recursive, recurrence relations for our two sequences when n ≥ 2 defined as above where U0 = 1, U1 = 0, V0 = 0, and V1 = 1:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U_{n}&=2V_{n-1}+U_{n-2}\\V_{n}&=U_{n-1}+V_{n-2}.\end{aligned}}}</annotation>

</semantics>

</math>

Since we have that for all integers m ≥ 0, the index-shifted generating functions satisfy<a href="#cite_note-23">[note 1]</a>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{m}G(z)=\sum _{n=m}^{\infty }g_{n-m}z^{n}\,,}</annotation>

</semantics>

</math>

we can use the initial conditions specified above and the previous two recurrence relations to see that we have the next two equations relating the generating functions for these sequences given by

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U(z)&=2zV(z)+z^{2}U(z)+1\\V(z)&=zU(z)+z^{2}V(z)={\frac {z}{1-z^{2}}}U(z),\end{aligned}}}</annotation>

</semantics>

</math>

which then implies by solving the system of equations (and this is the particular trick to our method here) that

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U(z)={\frac {1-z^{2}}{1-4z^{2}+z^{4}}}={\frac {1}{3-{\sqrt {3}}}}\cdot {\frac {1}{1-\left(2+{\sqrt {3}}\right)z^{2}}}+{\frac {1}{3+{\sqrt {3}}}}\cdot {\frac {1}{1-\left(2-{\sqrt {3}}\right)z^{2}}}.}</annotation>

</semantics>

</math>

Thus by performing algebraic simplifications to the sequence resulting from the second partial fractions expansions of the generating function in the previous equation, we find that U2n + 1 ≡ 0 and that

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<msup>

<mrow>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U_{2n}=\left\lceil {\frac {\left(2+{\sqrt {3}}\right)^{n}}{3-{\sqrt {3}}}}\right\rceil \,,}</annotation>

</semantics>

</math>

for all integers n ≥ 0. We also note that the same shifted generating function technique applied to the second-order <a href="Recurrence_relation" title="Recurrence relation">recurrence</a> for the <a href="Fibonacci_numbers" class="mw-redirect" title="Fibonacci numbers">Fibonacci numbers</a> is the prototypical example of using generating functions to solve recurrence relations in one variable already covered, or at least hinted at, in the subsection on <a href="Rational_functions" class="mw-redirect" title="Rational functions">rational functions</a> given above.

<div class="mw-heading mw-heading3"><h3 id="Convolution_(Cauchy_products)">Convolution (Cauchy products)</h3></div>

A discrete convolution of the terms in two formal power series turns a product of generating functions into a generating function enumerating a convolved sum of the original sequence terms (see <a href="Cauchy_product" title="Cauchy product">Cauchy product</a>).

<ol><li>Consider A(z) and B(z) are ordinary generating functions. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)=A(z)B(z)\Leftrightarrow [z^{n}]C(z)=\sum _{k=0}^{n}{a_{k}b_{n-k}}}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(z)=A(z)B(z)\Leftrightarrow [z^{n}]C(z)=\sum _{k=0}^{n}{a_{k}b_{n-k}}}</annotation>

</semantics>

</math></li>

<li>Consider A(z) and B(z) are exponential generating functions. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)=A(z)B(z)\Leftrightarrow \left[{\frac {z^{n}}{n!}}\right]C(z)=\sum _{k=0}^{n}{\binom {n}{k}}a_{k}b_{n-k}}">

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(z)=A(z)B(z)\Leftrightarrow \left[{\frac {z^{n}}{n!}}\right]C(z)=\sum _{k=0}^{n}{\binom {n}{k}}a_{k}b_{n-k}}</annotation>

</semantics>

</math></li>

<li>Consider the triply convolved sequence resulting from the product of three ordinary generating functions <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)=F(z)G(z)H(z)\Leftrightarrow [z^{n}]C(z)=\sum _{j+k+l=n}f_{j}g_{k}h_{l}}">

<msup>

</mrow>

</msup>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(z)=F(z)G(z)H(z)\Leftrightarrow [z^{n}]C(z)=\sum _{j+k+l=n}f_{j}g_{k}h_{l}}</annotation>

</semantics>

</math></li>

<li>Consider the m-fold convolution of a sequence with itself for some positive integer m ≥ 1 (see the example below for an application) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(z)=G(z)^{m}\Leftrightarrow [z^{n}]C(z)=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}g_{k_{1}}g_{k_{2}}\cdots g_{k_{m}}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(z)=G(z)^{m}\Leftrightarrow [z^{n}]C(z)=\sum _{k_{1}+k_{2}+\cdots +k_{m}=n}g_{k_{1}}g_{k_{2}}\cdots g_{k_{m}}}</annotation>

</semantics>

</math></li></ol>

Multiplication of generating functions, or convolution of their underlying sequences, can correspond to a notion of independent events in certain counting and probability scenarios. For example, if we adopt the notational convention that the <a href="Probability_generating_function" class="mw-redirect" title="Probability generating function">probability generating function</a>, or pgf, of a random variable Z is denoted by GZ(z), then we can show that for any two random variables <a href="#cite_note-24">[23]</a>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{X+Y}(z)=G_{X}(z)G_{Y}(z)\,,}</annotation>

</semantics>

</math>

if X and Y are independent.

<div class="mw-heading mw-heading4"><h4 id="Example:_The_money-changing_problem">Example: The money-changing problem</h4></div>

The number of ways to pay n ≥ 0 cents in coin denominations of values in the set {1, 5, 10, 25, 50} (i.e., in pennies, nickels, dimes, quarters, and half dollars, respectively), where we distinguish instances based upon the total number of each coin but not upon the order in which the coins are presented, is given by the ordinary generating function

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-z}}{\frac {1}{1-z^{5}}}{\frac {1}{1-z^{10}}}{\frac {1}{1-z^{25}}}{\frac {1}{1-z^{50}}}\,.}</annotation>

</semantics>

</math>

When we also distinguish based upon the order in which the coins are presented (e.g., one penny then one nickel is distinct from one nickel then one penny), the ordinary generating function is

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-z-z^{5}-z^{10}-z^{25}-z^{50}}}\,.}</annotation>

</semantics>

</math>

If we allow the n cents to be paid in coins of any positive integer denomination, we arrive at the <a href="Partition_function_(mathematics)" title="Partition function (mathematics)">partition function</a> ordinary generating function expanded by an infinite <a href="Q-Pochhammer_symbol" title="Q-Pochhammer symbol">q-Pochhammer symbol</a> product,

</mrow>

</mrow>

</munderover>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \prod _{n=1}^{\infty }\left(1-z^{n}\right)^{-1}\,.}</annotation>

</semantics>

</math>

When the order of the coins matters, the ordinary generating function is

<mfrac>

<mrow>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1-\sum _{n=1}^{\infty }z^{n}}}={\frac {1-z}{1-2z}}\,.}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Example:_Generating_function_for_the_Catalan_numbers">Example: Generating function for the Catalan numbers</h4></div>

An example where convolutions of generating functions are useful allows us to solve for a specific closed-form function representing the ordinary generating function for the <a href="Catalan_numbers" class="mw-redirect" title="Catalan numbers">Catalan numbers</a>, Cn. In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product x0 · x1 ·⋯· xn so that the order of multiplication is completely specified. For example, C2 = 2 which corresponds to the two expressions x0 · (x1 · x2) and (x0 · x1) · x2. It follows that the sequence satisfies a recurrence relation given by

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{n}=\sum _{k=0}^{n-1}C_{k}C_{n-1-k}+\delta _{n,0}=C_{0}C_{n-1}+C_{1}C_{n-2}+\cdots +C_{n-1}C_{0}+\delta _{n,0}\,,\quad n\geq 0\,,}</annotation>

</semantics>

</math>

and so has a corresponding convolved generating function, C(z), satisfying

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(z)=z\cdot C(z)^{2}+1\,.}</annotation>

</semantics>

</math>

Since C(0) = 1 ≠ ∞, we then arrive at a formula for this generating function given by

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(z)={\frac {1-{\sqrt {1-4z}}}{2z}}=\sum _{n=0}^{\infty }{\frac {1}{n+1}}{\binom {2n}{n}}z^{n}\,.}</annotation>

</semantics>

</math>

Note that the first equation implicitly defining C(z) above implies that

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C(z)={\frac {1}{1-z\cdot C(z)}}\,,}</annotation>

</semantics>

</math>

which then leads to another "simple" (of form) continued fraction expansion of this generating function.

<div class="mw-heading mw-heading4"><h4 id="Example:_Spanning_trees_of_fans_and_convolutions_of_convolutions">Example: Spanning trees of fans and convolutions of convolutions</h4></div>

A fan of order n is defined to be a graph on the vertices {0, 1, ..., n} with 2n − 1 edges connected according to the following rules: Vertex 0 is connected by a single edge to each of the other n vertices, and vertex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is connected by a single edge to the next vertex k + 1 for all 1 ≤ k < n.<a href="#cite_note-25">[24]</a> There is one fan of order one, three fans of order two, eight fans of order three, and so on. A <a href="Spanning_tree" title="Spanning tree">spanning tree</a> is a subgraph of a graph which contains all of the original vertices and which contains enough edges to make this subgraph connected, but not so many edges that there is a cycle in the subgraph. We ask how many spanning trees fn of a fan of order n are possible for each n ≥ 1.

As an observation, we may approach the question by counting the number of ways to join adjacent sets of vertices. For example, when n = 4, we have that f4 = 4 + 3 · 1 + 2 · 2 + 1 · 3 + 2 · 1 · 1 + 1 · 2 · 1 + 1 · 1 · 2 + 1 · 1 · 1 · 1 = 21, which is a sum over the m-fold convolutions of the sequence gn = n = [zn] ⁠z/(1 − z)2⁠ for m ≔ 1, 2, 3, 4. More generally, we may write a formula for this sequence as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{n}=\sum _{m>0}\sum _{\scriptstyle k_{1}+k_{2}+\cdots +k_{m}=n \atop \scriptstyle k_{1},k_{2},\ldots ,k_{m}>0}g_{k_{1}}g_{k_{2}}\cdots g_{k_{m}}\,,}">

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mfrac>

</mrow>

</munder>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{n}=\sum _{m>0}\sum _{\scriptstyle k_{1}+k_{2}+\cdots +k_{m}=n \atop \scriptstyle k_{1},k_{2},\ldots ,k_{m}>0}g_{k_{1}}g_{k_{2}}\cdots g_{k_{m}}\,,}</annotation>

</semantics>

</math>

from which we see that the ordinary generating function for this sequence is given by the next sum of convolutions as

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(z)=G(z)+G(z)^{2}+G(z)^{3}+\cdots ={\frac {G(z)}{1-G(z)}}={\frac {z}{(1-z)^{2}-z}}={\frac {z}{1-3z+z^{2}}}\,,}</annotation>

</semantics>

</math>

from which we are able to extract an exact formula for the sequence by taking the <a href="Partial_fraction_expansion" class="mw-redirect" title="Partial fraction expansion">partial fraction expansion</a> of the last generating function.

<div class="mw-heading mw-heading3"><h3 id="Implicit_generating_functions_and_the_Lagrange_inversion_formula">Implicit generating functions and the Lagrange inversion formula</h3></div>

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One often encounters generating functions specified by a functional equation, instead of an explicit specification. For example, the generating function T(z) for the number of binary trees on n nodes (leaves included) satisfies

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<annotation encoding="application/x-tex">{\displaystyle T(z)=z\left(1+T(z)^{2}\right)}</annotation>

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The <a href="Lagrange_inversion_theorem" title="Lagrange inversion theorem">Lagrange inversion theorem</a> is a tool used to explicitly evaluate solutions to such equations.

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Lagrange inversion formula—Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi (z)\in C[[z]]}">

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<annotation encoding="application/x-tex">{\textstyle \phi (z)\in C[[z]]}</annotation>

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</math><img src="./d15200790093e49d313e8d2b17f3c53f848e6939.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.565ex; height:2.843ex;" alt="{\textstyle \phi (z)\in C[[z]]}" loading="lazy"> be a formal power series with a non-zero constant term. Then the functional equation

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<annotation encoding="application/x-tex">{\displaystyle T(z)=z\phi (T(z))}</annotation>

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admits a unique solution in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T(z)\in C[[z]]}">

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<annotation encoding="application/x-tex">{\textstyle T(z)\in C[[z]]}</annotation>

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</math><img src="./d3dba50d4cc7281989b476fcb32d0a0c51552d8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.816ex; height:2.843ex;" alt="{\textstyle T(z)\in C[[z]]}" loading="lazy">, which satisfies

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<annotation encoding="application/x-tex">{\displaystyle [z^{n}]T(z)=[z^{n-1}]{\frac {1}{n}}(\phi (z))^{n}}</annotation>

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</math><img src="./2b79307844cb70ecaa92230ee6b4510ab2887fbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.479ex; height:5.176ex;" alt="{\displaystyle [z^{n}]T(z)=[z^{n-1}]{\frac {1}{n}}(\phi (z))^{n}}" loading="lazy">

where the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [z^{n}]F(z)}">

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<annotation encoding="application/x-tex">{\displaystyle [z^{n}]F(z)}</annotation>

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</math><img src="./a53406a968e38369b3deeb6092418b5e0c83d942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.241ex; height:2.843ex;" alt="{\displaystyle [z^{n}]F(z)}" loading="lazy"> returns the coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle z^{n}}</annotation>

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</math><img src="./5b1a8cdd7ee39054e510deeb38ee551cc7616ae1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.309ex; height:2.343ex;" alt="{\displaystyle z^{n}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(z)}">

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Applying the above theorem to our functional equation yields (with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \phi (z)=1+z^{2}}">

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<annotation encoding="application/x-tex">{\textstyle \phi (z)=1+z^{2}}</annotation>

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</math><img src="./01b92c2d6da6f1064598a44b3cb7203b1aaaaecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.529ex; height:3.009ex;" alt="{\textstyle \phi (z)=1+z^{2}}" loading="lazy">):

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<annotation encoding="application/x-tex">{\displaystyle [z^{n}]T(z)=[z^{n-1}]{\frac {1}{n}}(1+z^{2})^{n}}</annotation>

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Via the <a href="Binomial_theorem" title="Binomial theorem">binomial theorem</a> expansion, for even <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, the formula returns <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">. This is expected as one can prove that the number of leaves of a <a href="Binary_tree" title="Binary tree">binary tree</a> are one more than the number of its internal nodes, so the total sum should always be an odd number. For odd <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle [z^{n-1}]{\frac {1}{n}}(1+z^{2})^{n}={\frac {1}{n}}{\dbinom {n}{\frac {n+1}{2}}}}</annotation>

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The expression becomes much neater if we let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> be the number of internal nodes: Now the expression just becomes the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Introducing_a_free_parameter_(snake_oil_method)">Introducing a free parameter (snake oil method)</h3></div>

Sometimes the sum sn is complicated, and it is not always easy to evaluate. The "Free Parameter" method is another method (called "snake oil" by H. Wilf) to evaluate these sums.

Both methods discussed so far have n as limit in the summation. When n does not appear explicitly in the summation, we may consider n as a "free" parameter and treat sn as a coefficient of F(z) = Σ sn zn, change the order of the summations on n and k, and try to compute the inner sum.

For example, if we want to compute

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<annotation encoding="application/x-tex">{\displaystyle s_{n}=\sum _{k=0}^{\infty }{{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}}\,,\quad m,n\in \mathbb {N} _{0}\,,}</annotation>

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we can treat n as a "free" parameter, and set

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<annotation encoding="application/x-tex">{\displaystyle F(z)=\sum _{n=0}^{\infty }{\left(\sum _{k=0}^{\infty }{{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}}\right)}z^{n}\,.}</annotation>

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Interchanging summation ("snake oil") gives

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<annotation encoding="application/x-tex">{\displaystyle F(z)=\sum _{k=0}^{\infty }{{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}z^{-k}}\sum _{n=0}^{\infty }{{\binom {n+k}{m+2k}}z^{n+k}}\,.}</annotation>

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Now the inner sum is ⁠zm + 2k/(1 − z)m + 2k + 1⁠. Thus

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<mtext>where </mtext>

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<mtext>th Catalan number</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F(z)&={\frac {z^{m}}{(1-z)^{m+1}}}\sum _{k=0}^{\infty }{{\frac {1}{k+1}}{\binom {2k}{k}}\left({\frac {-z}{(1-z)^{2}}}\right)^{k}}\\[4px]&={\frac {z^{m}}{(1-z)^{m+1}}}\sum _{k=0}^{\infty }{C_{k}\left({\frac {-z}{(1-z)^{2}}}\right)^{k}}&{\text{where }}C_{k}=k{\text{th Catalan number}}\\[4px]&={\frac {z^{m}}{(1-z)^{m+1}}}{\frac {1-{\sqrt {1+{\frac {4z}{(1-z)^{2}}}}}}{\frac {-2z}{(1-z)^{2}}}}\\[4px]&={\frac {-z^{m-1}}{2(1-z)^{m-1}}}\left(1-{\frac {1+z}{1-z}}\right)\\[4px]&={\frac {z^{m}}{(1-z)^{m}}}=z{\frac {z^{m-1}}{(1-z)^{m}}}\,.\end{aligned}}}</annotation>

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Then we obtain

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<annotation encoding="application/x-tex">{\displaystyle s_{n}={\begin{cases}\displaystyle {\binom {n-1}{m-1}}&{\text{for }}m\geq 1\,,\\{}[n=0]&{\text{for }}m=0\,.\end{cases}}}</annotation>

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It is instructive to use the same method again for the sum, but this time take m as the free parameter instead of n. We thus set

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<annotation encoding="application/x-tex">{\displaystyle G(z)=\sum _{m=0}^{\infty }\left(\sum _{k=0}^{\infty }{\binom {n+k}{m+2k}}{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}\right)z^{m}\,.}</annotation>

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Interchanging summation ("snake oil") gives

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<annotation encoding="application/x-tex">{\displaystyle G(z)=\sum _{k=0}^{\infty }{\binom {2k}{k}}{\frac {(-1)^{k}}{k+1}}z^{-2k}\sum _{m=0}^{\infty }{\binom {n+k}{m+2k}}z^{m+2k}\,.}</annotation>

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Now the inner sum is (1 + z)n + k. Thus

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<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G(z)&=(1+z)^{n}\sum _{k=0}^{\infty }{\frac {1}{k+1}}{\binom {2k}{k}}\left({\frac {-(1+z)}{z^{2}}}\right)^{k}\\[4px]&=(1+z)^{n}\sum _{k=0}^{\infty }C_{k}\,\left({\frac {-(1+z)}{z^{2}}}\right)^{k}&{\text{where }}C_{k}=k{\text{th Catalan number}}\\[4px]&=(1+z)^{n}\,{\frac {1-{\sqrt {1+{\frac {4(1+z)}{z^{2}}}}}}{\frac {-2(1+z)}{z^{2}}}}\\[4px]&=(1+z)^{n}\,{\frac {z^{2}-z{\sqrt {z^{2}+4+4z}}}{-2(1+z)}}\\[4px]&=(1+z)^{n}\,{\frac {z^{2}-z(z+2)}{-2(1+z)}}\\[4px]&=(1+z)^{n}\,{\frac {-2z}{-2(1+z)}}=z(1+z)^{n-1}\,.\end{aligned}}}</annotation>

</semantics>

</math>

Thus we obtain

<msub>

</mrow>

</msub>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{n}=\left[z^{m}\right]z(1+z)^{n-1}=\left[z^{m-1}\right](1+z)^{n-1}={\binom {n-1}{m-1}}\,,}</annotation>

</semantics>

</math>

for m ≥ 1 as before.

<div class="mw-heading mw-heading3"><h3 id="Generating_functions_prove_congruences">Generating functions prove congruences</h3></div>

We say that two generating functions (power series) are congruent modulo m, written A(z) ≡ B(z) (mod m) if their coefficients are congruent modulo m for all n ≥ 0, i.e., an ≡ bn (mod m) for all relevant cases of the integers n (note that we need not assume that m is an integer here—it may very well be polynomial-valued in some indeterminate x, for example). If the "simpler" right-hand-side generating function, B(z), is a rational function of z, then the form of this sequence suggests that the sequence is <a href="Periodic_function" title="Periodic function">eventually periodic</a> modulo fixed particular cases of integer-valued m ≥ 2. For example, we can prove that the <a href="Euler_numbers" title="Euler numbers">Euler numbers</a>,

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \langle E_{n}\rangle =\langle 1,1,5,61,1385,\ldots \rangle \longmapsto \langle 1,1,2,1,2,1,2,\ldots \rangle {\pmod {3}}\,,}</annotation>

</semantics>

</math>

satisfy the following congruence modulo 3:<a href="#cite_note-26">[25]</a>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }E_{n}z^{n}={\frac {1-z^{2}}{1+z^{2}}}{\pmod {3}}\,.}</annotation>

</semantics>

</math>

One useful method of obtaining congruences for sequences enumerated by special generating functions modulo any integers (i.e., not only prime powers pk) is given in the section on continued fraction representations of (even non-convergent) ordinary generating functions by J-fractions above. We cite one particular result related to generating series expanded through a representation by continued fraction from Lando's Lectures on Generating Functions as follows:

Theorem: congruences for series generated by expansions of continued fractions—Suppose that the generating function A(z) is represented by an infinite <a href="Continued_fraction" title="Continued fraction">continued fraction</a> of the form

<mfrac>

<mrow>

</mpadded>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mpadded>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mpadded>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<mrow>

</mpadded>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A(z)={\cfrac {1}{1-c_{1}z-{\cfrac {p_{1}z^{2}}{1-c_{2}z-{\cfrac {p_{2}z^{2}}{1-c_{3}z-{\ddots }}}}}}}}</annotation>

</semantics>

</math>

and that Ap(z) denotes the pth convergent to this continued fraction expansion defined such that an = [zn] Ap(z) for all 0 ≤ n < 2p. Then:

<ol><li>the function Ap(z) is rational for all p ≥ 2 where we assume that one of divisibility criteria of p | p1, p1p2, p1p2p3 is met, that is, p | p1p2⋯pk for some k ≥ 1; and</li>

<li>if the integer p divides the product p1p2⋯pk, then we have A(z) ≡ Ak(z) (mod p).</li></ol>

</div>

Generating functions also have other uses in proving congruences for their coefficients. We cite the next two specific examples deriving special case congruences for the <a href="Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling numbers of the first kind</a> and for the <a href="Partition_function_(number_theory)" title="Partition function (number theory)">partition function p(n)</a> which show the versatility of generating functions in tackling problems involving <a href="Integer_sequences" class="mw-redirect" title="Integer sequences">integer sequences</a>.

<div class="mw-heading mw-heading4"><h4 id="The_Stirling_numbers_modulo_small_integers">The Stirling numbers modulo small integers</h4></div>

The <a href="Stirling_numbers_of_the_first_kind#Congruences" title="Stirling numbers of the first kind">main article</a> on the Stirling numbers generated by the finite products

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{n}(x):=\sum _{k=0}^{n}{\begin{bmatrix}n\\k\end{bmatrix}}x^{k}=x(x+1)(x+2)\cdots (x+n-1)\,,\quad n\geq 1\,,}</annotation>

</semantics>

</math>

provides an overview of the congruences for these numbers derived strictly from properties of their generating function as in Section 4.6 of Wilf's stock reference Generatingfunctionology.

We repeat the basic argument and notice that when reduces modulo 2, these finite product generating functions each satisfy

<msub>

</mrow>

</msub>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{n}(x)=[x(x+1)]\cdot [x(x+1)]\cdots =x^{\left\lceil {\frac {n}{2}}\right\rceil }(x+1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }\,,}</annotation>

</semantics>

</math>

which implies that the parity of these <a href="Stirling_numbers" class="mw-redirect" title="Stirling numbers">Stirling numbers</a> matches that of the binomial coefficient

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}n\\k\end{bmatrix}}\equiv {\binom {\left\lfloor {\frac {n}{2}}\right\rfloor }{k-\left\lceil {\frac {n}{2}}\right\rceil }}{\pmod {2}}\,,}</annotation>

</semantics>

</math>

and consequently shows that [n k] is even whenever k < ⌊ ⁠n/2⁠ ⌋.

Similarly, we can reduce the right-hand-side products defining the Stirling number generating functions modulo 3 to obtain slightly more complicated expressions providing that

<mtr>

<mtd>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

<mrow>

<mtr>

<mtd>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

<mrow>

<mtr>

<mtd>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

<msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\begin{bmatrix}n\\m\end{bmatrix}}&\equiv [x^{m}]\left(x^{\left\lceil {\frac {n}{3}}\right\rceil }(x+1)^{\left\lceil {\frac {n-1}{3}}\right\rceil }(x+2)^{\left\lfloor {\frac {n}{3}}\right\rfloor }\right)&&{\pmod {3}}\\&\equiv \sum _{k=0}^{m}{\begin{pmatrix}\left\lceil {\frac {n-1}{3}}\right\rceil \\k\end{pmatrix}}{\begin{pmatrix}\left\lfloor {\frac {n}{3}}\right\rfloor \\m-k-\left\lceil {\frac {n}{3}}\right\rceil \end{pmatrix}}\times 2^{\left\lceil {\frac {n}{3}}\right\rceil +\left\lfloor {\frac {n}{3}}\right\rfloor -(m-k)}&&{\pmod {3}}\,.\end{aligned}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading4"><h4 id="Congruences_for_the_partition_function">Congruences for the partition function</h4></div>

In this example, we pull in some of the machinery of infinite products whose power series expansions generate the expansions of many special functions and enumerate partition functions. In particular, we recall that the <a href="Partition_function_(number_theory)" title="Partition function (number theory)">partition function</a> p(n) is generated by the reciprocal infinite <a href="Q-Pochhammer_symbol" title="Q-Pochhammer symbol">q-Pochhammer symbol</a> product (or z-Pochhammer product as the case may be) given by

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<mfrac>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }p(n)z^{n}&={\frac {1}{\left(1-z\right)\left(1-z^{2}\right)\left(1-z^{3}\right)\cdots }}\\[4pt]&=1+z+2z^{2}+3z^{3}+5z^{4}+7z^{5}+11z^{6}+\cdots .\end{aligned}}}</annotation>

</semantics>

</math>

This partition function satisfies many known <a href="Ramanujan's_congruences" title="Ramanujan's congruences">congruence properties</a>, which notably include the following results though there are still many open questions about the forms of related integer congruences for the function:<a href="#cite_note-27">[26]</a>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}p(5m+4)&\equiv 0{\pmod {5}}\\p(7m+5)&\equiv 0{\pmod {7}}\\p(11m+6)&\equiv 0{\pmod {11}}\\p(25m+24)&\equiv 0{\pmod {5^{2}}}\,.\end{aligned}}}</annotation>

</semantics>

</math>

We show how to use generating functions and manipulations of congruences for formal power series to give a highly elementary proof of the first of these congruences listed above.

First, we observe that in the binomial coefficient generating function

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(1-z)^{5}}}=\sum _{i=0}^{\infty }{\binom {4+i}{4}}z^{i}\,,}</annotation>

</semantics>

</math>

all of the coefficients are divisible by 5 except for those which correspond to the powers 1, z5, z10, ... and moreover in those cases the remainder of the coefficient is 1 modulo 5. Thus,

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(1-z)^{5}}}\equiv {\frac {1}{1-z^{5}}}{\pmod {5}}\,,}</annotation>

</semantics>

</math>

or equivalently

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1-z^{5}}{(1-z)^{5}}}\equiv 1{\pmod {5}}\,.}</annotation>

</semantics>

</math>

It follows that

<mfrac>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<msup>

<mrow>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\left(1-z^{15}\right)\cdots }{\left((1-z)\left(1-z^{2}\right)\left(1-z^{3}\right)\cdots \right)^{5}}}\equiv 1{\pmod {5}}\,.}</annotation>

</semantics>

</math>

Using the infinite product expansions of

<mfrac>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

<mfrac>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<msup>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\cdot {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{\left(1-z\right)\left(1-z^{2}\right)\cdots }}=z\cdot \left((1-z)\left(1-z^{2}\right)\cdots \right)^{4}\times {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{\left(\left(1-z\right)\left(1-z^{2}\right)\cdots \right)^{5}}}\,,}</annotation>

</semantics>

</math>

it can be shown that the coefficient of z5m + 5 in z · ((1 − z)(1 − z2)⋯)4 is divisible by 5 for all m.<a href="#cite_note-28">[27]</a> Finally, since

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

<mfrac>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{n=1}^{\infty }p(n-1)z^{n}&={\frac {z}{(1-z)\left(1-z^{2}\right)\cdots }}\\[6px]&=z\cdot {\frac {\left(1-z^{5}\right)\left(1-z^{10}\right)\cdots }{(1-z)\left(1-z^{2}\right)\cdots }}\times \left(1+z^{5}+z^{10}+\cdots \right)\left(1+z^{10}+z^{20}+\cdots \right)\cdots \end{aligned}}}</annotation>

</semantics>

</math>

we may equate the coefficients of z5m + 5 in the previous equations to prove our desired congruence result, namely that p(5m + 4) ≡ 0 (mod 5) for all m ≥ 0.

<div class="mw-heading mw-heading3"><h3 id="Transformations_of_generating_functions">Transformations of generating functions</h3></div>

There are a number of transformations of generating functions that provide other applications (see the <a href="Generating_function_transformation" title="Generating function transformation">main article</a>). A transformation of a sequence's ordinary generating function (OGF) provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas involving a sequence OGF (see <a href="Generating_function_transformation#Integral_Transformations" title="Generating function transformation">integral transformations</a>) or weighted sums over the higher-order derivatives of these functions (see <a href="Generating_function_transformation#Derivative_Transformations" title="Generating function transformation">derivative transformations</a>).

Generating function transformations can come into play when we seek to express a generating function for the sums

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{n}:=\sum _{m=0}^{n}{\binom {n}{m}}C_{n,m}a_{m},}</annotation>

</semantics>

</math>

in the form of S(z) = g(z) A(f(z)) involving the original sequence generating function. For example, if the sums are

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{n}:=\sum _{k=0}^{\infty }{\binom {n+k}{m+2k}}a_{k}\,}</annotation>

</semantics>

</math>

then the generating function for the modified sum expressions is given by<a href="#cite_note-29">[28]</a>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S(z)={\frac {z^{m}}{(1-z)^{m+1}}}A\left({\frac {z}{(1-z)^{2}}}\right)}</annotation>

</semantics>

</math>

(see also the <a href="Binomial_transform" title="Binomial transform">binomial transform</a> and the <a href="Stirling_transform" title="Stirling transform">Stirling transform</a>).

There are also integral formulas for converting between a sequence's OGF, F(z), and its exponential generating function, or EGF, F̂(z), and vice versa given by

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</msubsup>

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}F(z)&=\int _{0}^{\infty }{\hat {F}}(tz)e^{-t}\,dt\,,\\[4px]{\hat {F}}(z)&={\frac {1}{2\pi }}\int _{-\pi }^{\pi }F\left(ze^{-i\vartheta }\right)e^{e^{i\vartheta }}\,d\vartheta \,,\end{aligned}}}</annotation>

</semantics>

</math>

provided that these integrals converge for appropriate values of z.

<div class="mw-heading mw-heading2"><h2 id="Tables_of_special_generating_functions">Tables of special generating functions</h2></div>

An initial listing of special mathematical series is found <a href="List_of_mathematical_series" title="List of mathematical series">here</a>. A number of useful and special sequence generating functions are found in Section 5.4 and 7.4 of Concrete Mathematics and in Section 2.5 of Wilf's Generatingfunctionology. Other special generating functions of note include the entries in the next table, which is by no means complete.<a href="#cite_note-30">[29]</a>

<th>Formal power series</th>

<th>Generating-function formula</th>

<th>Notes

</th></tr>

<tr>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\binom {m+n}{n}}\left(H_{n+m}-H_{m}\right)z^{n}}</annotation>

</semantics>

</math><img src="./460249a49b0156d4ba04a65570fa6864fad439ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.649ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\binom {m+n}{n}}\left(H_{n+m}-H_{m}\right)z^{n}}" loading="lazy"></td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(1-z)^{m+1}}}\ln {\frac {1}{1-z}}}</annotation>

</semantics>

</math><img src="./056701e71b4d4b68249106d4443855e91b26f1e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.153ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{(1-z)^{m+1}}}\ln {\frac {1}{1-z}}}" loading="lazy"></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{n}}</annotation>

</semantics>

</math><img src="./e63458b04288bbe116a9a8037dfae0b36b2c639a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.15ex; height:2.509ex;" alt="{\displaystyle H_{n}}" loading="lazy"> is a first-order <a href="Harmonic_number" title="Harmonic number">harmonic number</a>

</td></tr>

<tr>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }B_{n}{\frac {z^{n}}{n!}}}</annotation>

</semantics>

</math><img src="./c9902991620add9589e215cdef0b4556c97505c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.869ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }B_{n}{\frac {z^{n}}{n!}}}" loading="lazy"></td>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {z}{e^{z}-1}}}</annotation>

</semantics>

</math><img src="./555af2b62daebe76a76ee4bed8838038eab62de4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.924ex; height:4.843ex;" alt="{\displaystyle {\frac {z}{e^{z}-1}}}" loading="lazy"></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

</semantics>

</math><img src="./2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"> is a <a href="Bernoulli_number" title="Bernoulli number">Bernoulli number</a>

</td></tr>

<tr>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }F_{mn}z^{n}}</annotation>

</semantics>

</math><img src="./a1bc39d92a5ebcabfc16a6605293b928f5375e0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.207ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }F_{mn}z^{n}}" loading="lazy"></td>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {F_{m}z}{1-(F_{m-1}+F_{m+1})z+(-1)^{m}z^{2}}}}</annotation>

</semantics>

</math><img src="./0c42f991771fe4d3d4e02db77eee6453da014ef5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.557ex; height:6.009ex;" alt="{\displaystyle {\frac {F_{m}z}{1-(F_{m-1}+F_{m+1})z+(-1)^{m}z^{2}}}}" loading="lazy"></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{n}}</annotation>

</semantics>

</math><img src="./76cdf519c21deec43f984815e57e15d2dd3575d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.713ex; height:2.509ex;" alt="{\displaystyle F_{n}}" loading="lazy"> is a <a href="Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci number</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {Z} ^{+}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {Z} ^{+}}</annotation>

</semantics>

</math><img src="./3aa93f7da8db0f7cbbaaa8ce8ef18cb50d41e129.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.942ex; height:2.509ex;" alt="{\displaystyle m\in \mathbb {Z} ^{+}}" loading="lazy">

</td></tr>

<tr>

</mrow>

</mrow>

</munderover>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }\left\{{\begin{matrix}n\\m\end{matrix}}\right\}z^{n}}</annotation>

</semantics>

</math><img src="./499e8286cebad1e9c9d71e3c1fceb1e2e2509b64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.716ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }\left\{{\begin{matrix}n\\m\end{matrix}}\right\}z^{n}}" loading="lazy"></td>

<msup>

</mrow>

</msup>

<msup>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

</msup>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (z^{-1})^{\overline {-m}}={\frac {z^{m}}{(1-z)(1-2z)\cdots (1-mz)}}}</annotation>

</semantics>

</math><img src="./be5fa6e128cd1b24490e71d63cc69e6b5719695b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.615ex; height:6.009ex;" alt="{\displaystyle (z^{-1})^{\overline {-m}}={\frac {z^{m}}{(1-z)(1-2z)\cdots (1-mz)}}}" loading="lazy"></td>

<msup>

<mover>

</mover>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{\overline {n}}}</annotation>

</semantics>

</math><img src="./2c19902c6e1e5dca4e5265746c4b5e5d65a3c676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.641ex; height:2.843ex;" alt="{\displaystyle x^{\overline {n}}}" loading="lazy"> denotes the <a href="Rising_factorial" class="mw-redirect" title="Rising factorial">rising factorial</a>, or <a href="Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a> and some integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\geq 0}</annotation>

</semantics>

</math><img src="./b0d2d765e4cfd7adfbca9ae0e37e75a2811c0333.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.301ex; height:2.343ex;" alt="{\displaystyle m\geq 0}" loading="lazy">

</td></tr>

<tr>

</mrow>

</mrow>

</munderover>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }\left[{\begin{matrix}n\\m\end{matrix}}\right]z^{n}}</annotation>

</semantics>

</math><img src="./9ae0024b86ced7034b6a4501cee55be3bd269b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.685ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }\left[{\begin{matrix}n\\m\end{matrix}}\right]z^{n}}" loading="lazy"></td>

<msup>

<mover>

</mover>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{\overline {m}}=z(z+1)\cdots (z+m-1)}</annotation>

</semantics>

</math><img src="./6ffac8c46e8c9300ca686fba37f3020864d9398c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.224ex; height:3.343ex;" alt="{\displaystyle z^{\overline {m}}=z(z+1)\cdots (z+m-1)}" loading="lazy">

</td></tr>

<tr>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n-1}4^{n}(4^{n}-2)B_{2n}z^{2n}}{(2n)\cdot (2n)!}}}</annotation>

</semantics>

</math><img src="./3b94bcd4bb7e733f896ffbe28968e6f681e2e801.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.186ex; height:7.009ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n-1}4^{n}(4^{n}-2)B_{2n}z^{2n}}{(2n)\cdot (2n)!}}}" loading="lazy"></td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ln {\frac {\tan(z)}{z}}}</annotation>

</semantics>

</math><img src="./339b994d488077aa152a572185fe0d46aacc0c59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.42ex; height:5.676ex;" alt="{\displaystyle \ln {\frac {\tan(z)}{z}}}" loading="lazy">

</td></tr>

<tr>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<msup>

<mover>

</mover>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {(1/2)^{\overline {n}}z^{2n}}{(2n+1)\cdot n!}}}</annotation>

</semantics>

</math><img src="./62d503c6b4c5e2f1ae98983426c3d5668396010e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.668ex; height:7.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {(1/2)^{\overline {n}}z^{2n}}{(2n+1)\cdot n!}}}" loading="lazy"></td>

<msup>

</mrow>

</msup>

<mi>arcsin</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{-1}\arcsin(z)}</annotation>

</semantics>

</math><img src="./84a7fff022e479d3020969ada3a1a650f86cdf50.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.67ex; height:3.176ex;" alt="{\displaystyle z^{-1}\arcsin(z)}" loading="lazy">

</td></tr>

<tr>

</mrow>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }H_{n}^{(s)}z^{n}}</annotation>

</semantics>

</math><img src="./2a1f682985d26ce5ea119946a42c5c931162aad1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.437ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }H_{n}^{(s)}z^{n}}" loading="lazy"></td>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {Li} _{s}(z)}{1-z}}}</annotation>

</semantics>

</math><img src="./6f8205efc6fe99de91f2e2027ebca6a11ce0e2b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.837ex; height:5.843ex;" alt="{\displaystyle {\frac {\operatorname {Li} _{s}(z)}{1-z}}}" loading="lazy"></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Li} _{s}(z)}</annotation>

</semantics>

</math><img src="./4e650380250d97d59f35d1572402f4c952f05462.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6ex; height:2.843ex;" alt="{\displaystyle \operatorname {Li} _{s}(z)}" loading="lazy"> is the <a href="Polylogarithm" title="Polylogarithm">polylogarithm</a> function and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}^{(s)}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{n}^{(s)}}</annotation>

</semantics>

</math><img src="./9596b382ca909028c4c666377e7ae95bfdc5692b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.386ex; height:3.343ex;" alt="{\displaystyle H_{n}^{(s)}}" loading="lazy"> is a generalized <a href="Harmonic_number" title="Harmonic number">harmonic number</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Re (s)>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Re (s)>1}</annotation>

</semantics>

</math><img src="./9ea5968732ea35f67b36093364400e0fd8ca23bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle \Re (s)>1}" loading="lazy">

</td></tr>

<tr>

</mrow>

</mrow>

</munderover>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }n^{m}z^{n}}</annotation>

</semantics>

</math><img src="./608a8220eeb2e5daa17d10fcb79cb048e7c2e4a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:9.121ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }n^{m}z^{n}}" loading="lazy"></td>

</mrow>

</munder>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{0\leq j\leq m}\left\{{\begin{matrix}m\\j\end{matrix}}\right\}{\frac {j!\cdot z^{j}}{(1-z)^{j+1}}}}</annotation>

</semantics>

</math><img src="./1d54594b0404e150e3b06359eebc6d4065b16180.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.298ex; height:7.176ex;" alt="{\displaystyle \sum _{0\leq j\leq m}\left\{{\begin{matrix}m\\j\end{matrix}}\right\}{\frac {j!\cdot z^{j}}{(1-z)^{j+1}}}}" loading="lazy"></td>

<mrow>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{matrix}n\\m\end{matrix}}\right\}}</annotation>

</semantics>

</math><img src="./3ec8d5672e9a708baec85f4012902a9c0eaa396b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:6.278ex; height:6.176ex;" alt="{\displaystyle \left\{{\begin{matrix}n\\m\end{matrix}}\right\}}" loading="lazy"> is a <a href="Stirling_number_of_the_second_kind" class="mw-redirect" title="Stirling number of the second kind">Stirling number of the second kind</a> and where the individual terms in the expansion satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {z^{i}}{(1-z)^{i+1}}}=\sum _{k=0}^{i}{\binom {i}{k}}{\frac {(-1)^{k-i}}{(1-z)^{k+1}}}}">

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {z^{i}}{(1-z)^{i+1}}}=\sum _{k=0}^{i}{\binom {i}{k}}{\frac {(-1)^{k-i}}{(1-z)^{k+1}}}}</annotation>

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</math><img src="./a23c8a511fc71588941383c5d4aecab530db81c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.035ex; height:7.343ex;" alt="{\displaystyle {\frac {z^{i}}{(1-z)^{i+1}}}=\sum _{k=0}^{i}{\binom {i}{k}}{\frac {(-1)^{k-i}}{(1-z)^{k+1}}}}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{k<n}{\binom {n-k}{k}}{\frac {n}{n-k}}z^{k}}</annotation>

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</math><img src="./d8bf104229429a4479b7d0e37a096391946d3d5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.071ex; height:6.843ex;" alt="{\displaystyle \sum _{k<n}{\binom {n-k}{k}}{\frac {n}{n-k}}z^{k}}" loading="lazy"></td>

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<annotation encoding="application/x-tex">{\displaystyle \left({\frac {1+{\sqrt {1+4z}}}{2}}\right)^{n}+\left({\frac {1-{\sqrt {1+4z}}}{2}}\right)^{n}}</annotation>

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</math><img src="./7b2cd6a80be8e516d1298ac4cd6d8dce75085ae4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.176ex; height:6.509ex;" alt="{\displaystyle \left({\frac {1+{\sqrt {1+4z}}}{2}}\right)^{n}+\left({\frac {1-{\sqrt {1+4z}}}{2}}\right)^{n}}" loading="lazy"></td>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n_{1},\ldots ,n_{m}\geq 0}\min(n_{1},\ldots ,n_{m})z_{1}^{n_{1}}\cdots z_{m}^{n_{m}}}</annotation>

</semantics>

</math><img src="./2f22807cefac95870e2cd2a376fd8eef6b05f387.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.131ex; height:5.843ex;" alt="{\displaystyle \sum _{n_{1},\ldots ,n_{m}\geq 0}\min(n_{1},\ldots ,n_{m})z_{1}^{n_{1}}\cdots z_{m}^{n_{m}}}" loading="lazy"></td>

<mfrac>

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {z_{1}\cdots z_{m}}{(1-z_{1})\cdots (1-z_{m})(1-z_{1}\cdots z_{m})}}}</annotation>

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</math><img src="./6e96e44a974e19d9615283d458a93ad128425b14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.051ex; height:5.509ex;" alt="{\displaystyle {\frac {z_{1}\cdots z_{m}}{(1-z_{1})\cdots (1-z_{m})(1-z_{1}\cdots z_{m})}}}" loading="lazy"></td>

<td>The two-variable case is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(w,z):=\sum _{m,n\geq 0}\min(m,n)w^{m}z^{n}={\frac {wz}{(1-w)(1-z)(1-wz)}}}">

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<annotation encoding="application/x-tex">{\displaystyle M(w,z):=\sum _{m,n\geq 0}\min(m,n)w^{m}z^{n}={\frac {wz}{(1-w)(1-z)(1-wz)}}}</annotation>

</semantics>

</math><img src="./0b961c9aa7d3930988f1a74d115cac4cbc3f8d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:59.834ex; height:6.176ex;" alt="{\displaystyle M(w,z):=\sum _{m,n\geq 0}\min(m,n)w^{m}z^{n}={\frac {wz}{(1-w)(1-z)(1-wz)}}}" loading="lazy">

</td></tr>

<tr>

</mrow>

</mrow>

</munderover>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\binom {s}{n}}z^{n}}</annotation>

</semantics>

</math><img src="./39b94893b1e51a1051708b62dba9eb531a75b72c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.867ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\binom {s}{n}}z^{n}}" loading="lazy"></td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1+z)^{s}}</annotation>

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</math><img src="./b1cf46051ebfea509a48b43fe2899437830cd2dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.904ex; height:2.843ex;" alt="{\displaystyle (1+z)^{s}}" loading="lazy"></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in \mathbb {C} }</annotation>

</semantics>

</math><img src="./f13da7c2bac9dd6324ac83093fb42f3a9e86fbd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.609ex; height:2.176ex;" alt="{\displaystyle s\in \mathbb {C} }" loading="lazy">

</td></tr>

<tr>

</mrow>

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</mrow>

</mfrac>

</mrow>

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\binom {n}{k}}z^{n}}</annotation>

</semantics>

</math><img src="./3c1ae8325fa132fc63f45ba302208bc8a5d1776e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.867ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\binom {n}{k}}z^{n}}" loading="lazy"></td>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {z^{k}}{(1-z)^{k+1}}}}</annotation>

</semantics>

</math><img src="./e76587709171c622d8c4f22a34566cac1c8930ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:10.926ex; height:6.509ex;" alt="{\displaystyle {\frac {z^{k}}{(1-z)^{k+1}}}}" loading="lazy"></td>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation>

</semantics>

</math><img src="./2a5bc4b7383031ba693b7433198ead7170954c1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }" loading="lazy">

</td></tr>

<tr>

</mrow>

</mrow>

</munderover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }\log {(n)}z^{n}}</annotation>

</semantics>

</math><img src="./26bd8cdaf9a0c810a87a9ca2b63b9f480e4f39a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.614ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }\log {(n)}z^{n}}" loading="lazy"></td>

<msub>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left.-{\frac {\partial }{\partial s}}\operatorname {{Li}_{s}(z)} \right|_{s=0}}</annotation>

</semantics>

</math><img src="./2c12f2bd1651494d6861b867c0316335a02b032a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.012ex; height:6.009ex;" alt="{\displaystyle \left.-{\frac {\partial }{\partial s}}\operatorname {{Li}_{s}(z)} \right|_{s=0}}" loading="lazy">

</td></tr></tbody></table>

<ul><li><a href="Moment-generating_function" title="Moment-generating function">Moment-generating function</a></li>

<li><a href="Probability-generating_function" title="Probability-generating function">Probability-generating function</a></li>

<li><a href="Generating_function_transformation" title="Generating function transformation">Generating function transformation</a></li>

<li><a href="Stanley's_reciprocity_theorem" title="Stanley's reciprocity theorem">Stanley's reciprocity theorem</a></li>

<li><a href="Integer_partition" title="Integer partition">Integer partition</a></li>

<li><a href="Combinatorial_principles" title="Combinatorial principles">Combinatorial principles</a></li>

<li><a href="Cyclic_sieving" title="Cyclic sieving">Cyclic sieving</a></li>

<li><a href="Z-transform" title="Z-transform">Z-transform</a></li>

<li><a href="Umbral_calculus" title="Umbral calculus">Umbral calculus</a></li>

<li><a href="Coins_in_a_fountain" title="Coins in a fountain">Coins in a fountain</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-23"><a href="#cite_ref-23">^</a> Incidentally, we also have a corresponding formula when m < 0 given by

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }g_{n+m}z^{n}={\frac {G(z)-g_{0}-g_{1}z-\cdots -g_{m-1}z^{m-1}}{z^{m}}}\,.}</annotation>

</semantics>

</math>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">^</a> This alternative term can already be found in E.N. Gilbert (1956), "Enumeration of Labeled graphs", <a href="Canadian_Journal_of_Mathematics" title="Canadian Journal of Mathematics">Canadian Journal of Mathematics</a> 3, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=x34z99fCRbsC&dq=%22generating+series%22&pg=PA407">p. 405–411</a>, but its use is rare before the year 2000; since then it appears to be increasing.

</li>

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</style><cite id="CITEREFKnuth1997" class="citation book cs1"><a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a> (1997). "§1.2.9 Generating Functions". Fundamental Algorithms. <a href="The_Art_of_Computer_Programming" title="The Art of Computer Programming">The Art of Computer Programming</a>. Vol. 1 (3rd ed.). Addison-Wesley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-89683-4</bdi>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFFlajoletSedgewick2009">Flajolet & Sedgewick 2009</a>, p. 95

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/140418">"Lambert series identity"</a>. Math Overflow. 2017.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFApostol1976" class="citation cs2"><a href="Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90163-3</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929">0434929</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0335.10001">0335.10001</a></cite> pp.42–43

</li>

<li id="cite_note-W56-6"><a href="#cite_ref-W56_6-0">^</a> <a href="#CITEREFWilf1994">Wilf 1994</a>, p. 56

</li>

<li id="cite_note-W59-7"><a href="#cite_ref-W59_7-0">^</a> <a href="#CITEREFWilf1994">Wilf 1994</a>, p. 59

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFHardyWrightHeath-BrownSilverman2008" class="citation book cs1">Hardy, G.H.; Wright, E.M.; Heath-Brown, D.R; Silverman, J.H. (2008). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00ghha_922">An Introduction to the Theory of Numbers</a> (6th ed.). Oxford University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00ghha_922/page/n357">339</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780199219858</bdi>.</cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFKnuth1992" class="citation journal cs1">Knuth, D. E. (1992). "Convolution Polynomials". Mathematica J. 2: 67–78. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9207221">math/9207221</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992math......7221K">1992math......7221K</a>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFSpivey2007" class="citation journal cs1">Spivey, Michael Z. (2007). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.disc.2007.03.052">"Combinatorial Sums and Finite Differences"</a>. Discrete Math. 307 (24): 3130–3146. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.disc.2007.03.052">10.1016/j.disc.2007.03.052</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2370116">2370116</a>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFMathar2012" class="citation arxiv cs1">Mathar, R. J. (2012). "Yet another table of integrals". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1207.5845">1207.5845</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CA">math.CA</a>].</cite> v4 eq. (0.4)

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, Table 265 in §6.1 for finite sum identities involving the Stirling number triangles.

</li>

<li id="cite_note-GFLECT-13"><a href="#cite_ref-GFLECT_13-0">^</a> <a href="#CITEREFLando2003">Lando 2003</a>, §2.4

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> Example from <cite id="CITEREFStanleyFomin1997" class="citation book cs1">Stanley, Richard P.; Fomin, Sergey (1997). "§6.3". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=zg5wDqT6T-UC">Enumerative Combinatorics: Volume 2</a>. Cambridge Studies in Advanced Mathematics. Vol. 62. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-78987-5</bdi>.</cite>

</li>

<li id="cite_note-TAOCPV1-15"><a href="#cite_ref-TAOCPV1_15-0">^</a> <a href="#CITEREFKnuth1997">Knuth 1997</a>, §1.2.9

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> Solution to <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, p. 569, exercise 7.36

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFFlajoletSedgewick2009">Flajolet & Sedgewick 2009</a>, §B.4

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFSchneider2007" class="citation journal cs1">Schneider, C. (2007). <a rel="nofollow" class="external text" href="http://www.emis.de/journals/SLC/wpapers/s56schneider.html">"Symbolic Summation Assists Combinatorics"</a>. Sém. Lothar. Combin. 56: 1–36.</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> See the usage of these terms in <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, §7.4 on special sequence generating functions.

</li>

<li id="cite_note-Good_1986-20"><a href="#cite_ref-Good_1986_20-0">^</a> <cite id="CITEREFGood1986" class="citation journal cs1">Good, I. J. (1986). <a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176343649">"On applications of symmetric Dirichlet distributions and their mixtures to contingency tables"</a>. <a href="Annals_of_Statistics" title="Annals of Statistics">Annals of Statistics</a>. 4 (6): 1159–1189. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1214%2Faos%2F1176343649">10.1214/aos/1176343649</a>.</cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> For more complete information on the properties of J-fractions see:

<ul><li><cite id="CITEREFFlajolet1980" class="citation journal cs1">Flajolet, P. (1980). <a rel="nofollow" class="external text" href="http://algo.inria.fr/flajolet/Publications/Flajolet80b.pdf">"Combinatorial aspects of continued fractions"</a> (PDF). Discrete Mathematics. 32 (2): 125–161. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0012-365X%2880%2990050-3">10.1016/0012-365X(80)90050-3</a>.</cite></li>

<li><cite id="CITEREFWall2018" class="citation book cs1">Wall, H.S. (2018) [1948]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=86ReDwAAQBAJ&pg=PR7">Analytic Theory of Continued Fractions</a>. Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-83044-5</bdi>.</cite></li></ul>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> See the following articles:

<ul><li><cite id="CITEREFSchmidt2016" class="citation arxiv cs1">Schmidt, Maxie D. (2016). "Continued Fractions for Square Series Generating Functions". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1612.02778">1612.02778</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.NT">math.NT</a>].</cite></li>

<li><cite id="CITEREFSchmidt2017" class="citation journal cs1">— (2017). <a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Schmidt/schmidt14.html">"Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions"</a>. Journal of Integer Sequences. 20. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1610.09691">1610.09691</a>. 17.3.4.</cite></li>

<li><cite id="CITEREFSchmidt2017" class="citation arxiv cs1">— (2017). "Jacobi-Type Continued Fractions and Congruences for Binomial Coefficients Modulo Integers h ≥ 2". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1702.01374">1702.01374</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.CO">math.CO</a>].</cite></li></ul>

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, §8.3

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, Example 6 in §7.3 for another method and the complete setup of this problem using generating functions. This more "convoluted" approach is given in Section 7.5 of the same reference.

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFLando2003">Lando 2003</a>, §5

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFHardyWrightHeath-BrownSilverman2008">Hardy et al. 2008</a>, §19.12

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> <cite id="CITEREFHardyWright" class="citation book cs1">Hardy, G.H.; Wright, E.M. An Introduction to the Theory of Numbers.</cite> p.288, Th.361

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <a href="#CITEREFGrahamKnuthPatashnik1994">Graham, Knuth & Patashnik 1994</a>, p. 535, exercise 5.71

</li>

<li id="cite_note-30"><a href="#cite_ref-30">^</a> See also the 1031 Generating Functions found in <cite id="CITEREFPlouffe1992" class="citation thesis cs1 cs1-prop-foreign-lang-source">Plouffe, Simon (1992). Approximations de séries génératrices et quelques conjectures [Approximations of generating functions and a few conjectures] (Masters) (in French). Université du Québec à Montréal. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0911.4975">0911.4975</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

<ul><li><cite id="CITEREFAigner2007" class="citation book cs1">Aigner, Martin (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pPEJcu93dzAC">A Course in Enumeration</a>. Graduate Texts in Mathematics. Vol. 238. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-39035-0</bdi>.</cite></li>

<li><cite id="CITEREFDoubiletRotaStanley1972" class="citation journal cs1">Doubilet, Peter; <a href="Gian-Carlo_Rota" title="Gian-Carlo Rota">Rota, Gian-Carlo</a>; <a href="Richard_P._Stanley" title="Richard P. Stanley">Stanley, Richard</a> (1972). <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.bsmsp/1200514223">"On the foundations of combinatorial theory. VI. The idea of generating function"</a>. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. 2: 267–318. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0267.05002">0267.05002</a>.</cite> Reprinted in <cite id="CITEREFRota1975" class="citation book cs1"><a href="Gian-Carlo_Rota" title="Gian-Carlo Rota">Rota, Gian-Carlo</a> (1975). "3. The idea of generating function". Finite Operator Calculus. With the collaboration of P. Doubilet, C. Greene, D. Kahaner, <a href="Andrew_Odlyzko" title="Andrew Odlyzko">A. Odlyzko</a> and <a href="Richard_P._Stanley" title="Richard P. Stanley">R. Stanley</a>. Academic Press. pp. 83–134. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-596650-4</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0328.05007">0328.05007</a>.</cite></li>

<li><cite id="CITEREFFlajoletSedgewick2009" class="citation book cs1"><a href="Philippe_Flajolet" title="Philippe Flajolet">Flajolet, Philippe</a>; <a href="Robert_Sedgewick_(computer_scientist)" title="Robert Sedgewick (computer scientist)">Sedgewick, Robert</a> (2009). <a href="Analytic_Combinatorics" class="mw-redirect" title="Analytic Combinatorics">Analytic Combinatorics</a>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-89806-5</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1165.05001">1165.05001</a>.</cite></li>

<li><cite id="CITEREFGouldenJackson2004" class="citation book cs1">Goulden, Ian P.; <a href="David_M._Jackson" title="David M. Jackson">Jackson, David M.</a> (2004). Combinatorial Enumeration. <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0486435978</bdi>.</cite></li>

<li><cite id="CITEREFGrahamKnuthPatashnik1994" class="citation book cs1"><a href="Ronald_Graham" title="Ronald Graham">Graham, Ronald L.</a>; <a href="Donald_Knuth" title="Donald Knuth">Knuth, Donald E.</a>; <a href="Oren_Patashnik" title="Oren Patashnik">Patashnik, Oren</a> (1994). "Chapter 7: Generating Functions". <a href="Concrete_Mathematics" title="Concrete Mathematics">Concrete Mathematics. A foundation for computer science</a> (2nd ed.). Addison-Wesley. pp. 320–380. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-55802-5</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0836.00001">0836.00001</a>.</cite></li>

<li><cite id="CITEREFLando2003" class="citation book cs1">Lando, Sergei K. (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=A6_4AwAAQBAJ">Lectures on Generating Functions</a>. American Mathematical Society. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-3481-7</bdi>.</cite></li>

<li><cite id="CITEREFWilf1994" class="citation book cs1"><a href="Herbert_Wilf" title="Herbert Wilf">Wilf, Herbert S.</a> (1994). <a rel="nofollow" class="external text" href="http://www.math.upenn.edu/%7Ewilf/DownldGF.html">Generatingfunctionology</a> (2nd ed.). Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-12-751956-4</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0831.05001">0831.05001</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://garsia.math.yorku.ca/~zabrocki/MMM1/MMM1Intro2OGFs.pdf">"Introduction To Ordinary Generating Functions"</a> by Mike Zabrocki, York University, Mathematics and Statistics</li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Generating_function">"Generating function"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/ctk/GeneratingFunctions.shtml">Generating Functions, Power Indices and Coin Change</a> at <a href="Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a></li>

<li><a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/GeneratingFunctions/">"Generating Functions"</a> by <a href="Ed_Pegg_Jr." title="Ed Pegg Jr.">Ed Pegg Jr.</a>, <a href="Wolfram_Demonstrations_Project" title="Wolfram Demonstrations Project">Wolfram Demonstrations Project</a>, 2007.</li></ul>

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