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<title>Generalized hypergeometric function</title>

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Generalized hypergeometric function

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</style><div role="note" class="hatnote navigation-not-searchable">For other generalizations of the hypergeometric function, see <a href="Hypergeometric_function" title="Hypergeometric function">hypergeometric function</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="General_hypergeometric_function" title="General hypergeometric function">general hypergeometric function</a>.</div><div role="note" class="hatnote navigation-not-searchable">"PFq" redirects here. For other uses, see <a href="PFQ_(disambiguation)" class="mw-redirect mw-disambig" title="PFQ (disambiguation)">PFQ (disambiguation)</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, a generalized hypergeometric series is a <a href="Power_series" title="Power series">power series</a> in which the ratio of successive <a href="Coefficient" title="Coefficient">coefficients</a> indexed by n is a <a href="Rational_function" title="Rational function">rational function</a> of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by <a href="Analytic_continuation" title="Analytic continuation">analytic continuation</a>. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the <a href="Gaussian_hypergeometric_series" class="mw-redirect" title="Gaussian hypergeometric series">Gaussian hypergeometric series</a>. Generalized hypergeometric functions include the (Gaussian) <a href="Hypergeometric_function" title="Hypergeometric function">hypergeometric function</a> and the <a href="Confluent_hypergeometric_function" title="Confluent hypergeometric function">confluent hypergeometric function</a> as special cases, which in turn have many particular <a href="Special_functions" title="Special functions">special functions</a> as special cases, such as <a href="Elementary_functions" class="mw-redirect" title="Elementary functions">elementary functions</a>, <a href="Bessel_function" title="Bessel function">Bessel functions</a>, and the <a href="Orthogonal_polynomials" title="Orthogonal polynomials">classical orthogonal polynomials</a>.

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

A hypergeometric series is formally defined as a <a href="Power_series" title="Power series">power series</a>

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<annotation encoding="application/x-tex">{\displaystyle \beta _{0}+\beta _{1}z+\beta _{2}z^{2}+\dots =\sum _{n\geqslant 0}\beta _{n}z^{n}}</annotation>

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</math><img src="./35f931c3d0d72eb81b0506e706a548d10944f2aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.271ex; height:5.676ex;" alt="{\displaystyle \beta _{0}+\beta _{1}z+\beta _{2}z^{2}+\dots =\sum _{n\geqslant 0}\beta _{n}z^{n}}" loading="lazy"></dd></dl>

in which the ratio of successive coefficients is a <a href="Rational_function" title="Rational function">rational function</a> of n. That is,

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\beta _{n+1}}{\beta _{n}}}={\frac {A(n)}{B(n)}}}</annotation>

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</math><img src="./27da9837637abf5cdbc2a46d8e82c07897243c34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.373ex; height:6.509ex;" alt="{\displaystyle {\frac {\beta _{n+1}}{\beta _{n}}}={\frac {A(n)}{B(n)}}}" loading="lazy"></dd></dl>

where A(n) and B(n) are <a href="Polynomial" title="Polynomial">polynomials</a> in n.

For example, in the case of the series for the <a href="Exponential_function" title="Exponential function">exponential function</a>,

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

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</math><img src="./34987b7e8a09c47d14f1102e347884443c5ed693.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.888ex; height:5.843ex;" alt="{\displaystyle 1+{\frac {z}{1!}}+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+\cdots ,}" loading="lazy"></dd></dl>

we have:

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<annotation encoding="application/x-tex">{\displaystyle \beta _{n}={\frac {1}{n!}},\qquad {\frac {\beta _{n+1}}{\beta _{n}}}={\frac {1}{n+1}}.}</annotation>

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</math><img src="./f2806c7d1eaab8654790993e2266797b32f8f923.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.639ex; height:5.843ex;" alt="{\displaystyle \beta _{n}={\frac {1}{n!}},\qquad {\frac {\beta _{n+1}}{\beta _{n}}}={\frac {1}{n+1}}.}" loading="lazy"></dd></dl>

So this satisfies the definition with A(n) = 1 and B(n) = n + 1.

It is customary to factor out the leading term, so β0 is assumed to be 1. The polynomials can be factored into linear factors of the form (aj + n) and (bk + n) respectively, where the aj and bk are <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>.

For historical reasons, it is assumed that (1 + n) is a factor of B. If this is not already the case then both A and B can be multiplied by this factor; the factor cancels so the terms are unchanged and there is no loss of generality.

The ratio between consecutive coefficients now has the form

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<annotation encoding="application/x-tex">{\displaystyle {\frac {c(a_{1}+n)\cdots (a_{p}+n)}{d(b_{1}+n)\cdots (b_{q}+n)(1+n)}}}</annotation>

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</math><img src="./ee80e4b0e0c70f458e08b37c61a6200f8a24e565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.883ex; height:6.509ex;" alt="{\displaystyle {\frac {c(a_{1}+n)\cdots (a_{p}+n)}{d(b_{1}+n)\cdots (b_{q}+n)(1+n)}}}" loading="lazy">,</dd></dl>

where c and d are the leading coefficients of A and B. The series then has the form

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<annotation encoding="application/x-tex">{\displaystyle 1+{\frac {a_{1}\cdots a_{p}}{b_{1}\cdots b_{q}\cdot 1}}{\frac {cz}{d}}+{\frac {a_{1}\cdots a_{p}}{b_{1}\cdots b_{q}\cdot 1}}{\frac {(a_{1}+1)\cdots (a_{p}+1)}{(b_{1}+1)\cdots (b_{q}+1)\cdot 2}}\left({\frac {cz}{d}}\right)^{2}+\cdots }</annotation>

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</math><img src="./e88f20747e1e405854da21e25df8a30dde5a9947.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:67.362ex; height:6.509ex;" alt="{\displaystyle 1+{\frac {a_{1}\cdots a_{p}}{b_{1}\cdots b_{q}\cdot 1}}{\frac {cz}{d}}+{\frac {a_{1}\cdots a_{p}}{b_{1}\cdots b_{q}\cdot 1}}{\frac {(a_{1}+1)\cdots (a_{p}+1)}{(b_{1}+1)\cdots (b_{q}+1)\cdot 2}}\left({\frac {cz}{d}}\right)^{2}+\cdots }" loading="lazy">,</dd></dl>

or, by scaling z by the appropriate factor and rearranging,

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<annotation encoding="application/x-tex">{\displaystyle 1+{\frac {a_{1}\cdots a_{p}}{b_{1}\cdots b_{q}}}{\frac {z}{1!}}+{\frac {a_{1}(a_{1}+1)\cdots a_{p}(a_{p}+1)}{b_{1}(b_{1}+1)\cdots b_{q}(b_{q}+1)}}{\frac {z^{2}}{2!}}+\cdots }</annotation>

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</math><img src="./d495854ee2599146d63cb1162fdeced678688faa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:52.044ex; height:6.509ex;" alt="{\displaystyle 1+{\frac {a_{1}\cdots a_{p}}{b_{1}\cdots b_{q}}}{\frac {z}{1!}}+{\frac {a_{1}(a_{1}+1)\cdots a_{p}(a_{p}+1)}{b_{1}(b_{1}+1)\cdots b_{q}(b_{q}+1)}}{\frac {z^{2}}{2!}}+\cdots }" loading="lazy">.</dd></dl>

This has the form of an <a href="Generating_function" title="Generating function">exponential generating function</a>. This series is usually denoted by

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<annotation encoding="application/x-tex">{\displaystyle {}_{p}F_{q}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};z)}</annotation>

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</math><img src="./f0118fa5b2ea5391fac420be3691e37efe2b3c6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.063ex; width:27.538ex; height:3.009ex;" alt="{\displaystyle {}_{p}F_{q}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};z)}" loading="lazy"></dd></dl>

or

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<annotation encoding="application/x-tex">{\displaystyle \,{}_{p}F_{q}\left[{\begin{matrix}a_{1}&a_{2}&\cdots &a_{p}\\b_{1}&b_{2}&\cdots &b_{q}\end{matrix}};z\right].}</annotation>

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</math><img src="./926ccb7c81b723d4bfbae1c54bb99e7eff703932.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.227ex; height:6.509ex;" alt="{\displaystyle \,{}_{p}F_{q}\left[{\begin{matrix}a_{1}&a_{2}&\cdots &a_{p}\\b_{1}&b_{2}&\cdots &b_{q}\end{matrix}};z\right].}" loading="lazy"></dd></dl>

Using the rising factorial or <a href="Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(a)_{0}&=1,\\(a)_{n}&=a(a+1)(a+2)\cdots (a+n-1)={\frac {\Gamma (a+n)}{\Gamma (a)}},&&n\geq 1,\end{aligned}}}</annotation>

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</math><img src="./62155e69a1448c06123139c36933b3a924b1be0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:62.452ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}(a)_{0}&=1,\\(a)_{n}&=a(a+1)(a+2)\cdots (a+n-1)={\frac {\Gamma (a+n)}{\Gamma (a)}},&&n\geq 1,\end{aligned}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (x)}">

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

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</math><img src="./ec077ba0bdbf87c0d66173bc4d98598fe582ac37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.592ex; height:2.843ex;" alt="{\displaystyle \Gamma (x)}" loading="lazy"> represents the <a href="Gamma_function" title="Gamma function">gamma function</a>, this can be written

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<annotation encoding="application/x-tex">{\displaystyle \,{}_{p}F_{q}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};z)=\sum _{n=0}^{\infty }{\frac {(a_{1})_{n}\cdots (a_{p})_{n}}{(b_{1})_{n}\cdots (b_{q})_{n}}}\,{\frac {z^{n}}{n!}}={\frac {\Gamma (b_{1})\cdots \Gamma (b_{q})}{\Gamma (a_{1})\cdots \Gamma (a_{p})}}\sum _{n=0}^{\infty }{\frac {\Gamma (n+a_{1})\cdots \Gamma (n+a_{p})}{\Gamma (n+b_{1})\cdots \Gamma (n+b_{q})}}{\frac {z^{n}}{n!}}.}</annotation>

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</math><img src="./fd844951373c25f1192062886944b306dec857ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:103.548ex; height:6.843ex;" alt="{\displaystyle \,{}_{p}F_{q}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};z)=\sum _{n=0}^{\infty }{\frac {(a_{1})_{n}\cdots (a_{p})_{n}}{(b_{1})_{n}\cdots (b_{q})_{n}}}\,{\frac {z^{n}}{n!}}={\frac {\Gamma (b_{1})\cdots \Gamma (b_{q})}{\Gamma (a_{1})\cdots \Gamma (a_{p})}}\sum _{n=0}^{\infty }{\frac {\Gamma (n+a_{1})\cdots \Gamma (n+a_{p})}{\Gamma (n+b_{1})\cdots \Gamma (n+b_{q})}}{\frac {z^{n}}{n!}}.}" loading="lazy"></dd></dl>

(Note that this use of the Pochhammer symbol is not standard; however it is the standard usage in this context.)

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

When all the terms of the series are defined and it has a non-zero <a href="Radius_of_convergence" title="Radius of convergence">radius of convergence</a>, then the series defines an <a href="Analytic_function" title="Analytic function">analytic function</a>. Such a function, and its <a href="Analytic_continuation" title="Analytic continuation">analytic continuations</a>, is called the hypergeometric function.

The case when the radius of convergence is 0 yields many interesting series in mathematics, for example the <a href="Incomplete_gamma_function" title="Incomplete gamma function">incomplete gamma function</a> has the <a href="Asymptotic_expansion" title="Asymptotic expansion">asymptotic expansion</a>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma (a,z)\sim z^{a-1}e^{-z}\left(1+{\frac {a-1}{z}}+{\frac {(a-1)(a-2)}{z^{2}}}+\cdots \right)}</annotation>

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</math><img src="./5cfe08bc7795c2dc4a4086d7ed5e8633b042d496.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.573ex; height:6.343ex;" alt="{\displaystyle \Gamma (a,z)\sim z^{a-1}e^{-z}\left(1+{\frac {a-1}{z}}+{\frac {(a-1)(a-2)}{z^{2}}}+\cdots \right)}" loading="lazy"></dd></dl>

which could be written za−1e−z 2F0(1−a,1;;−z−1). However, the use of the term hypergeometric series is usually restricted to the case where the series defines an actual analytic function.

The ordinary hypergeometric series should not be confused with the <a href="Basic_hypergeometric_series" title="Basic hypergeometric series">basic hypergeometric series</a>, which, despite its name, is a rather more complicated and recondite series. The "basic" series is the <a href="Q-analog" title="Q-analog">q-analog</a> of the ordinary hypergeometric series. There are several such generalizations of the ordinary hypergeometric series, including the ones coming from <a href="Zonal_spherical_function" title="Zonal spherical function">zonal spherical functions</a> on <a href="Symmetric_space" title="Symmetric space">Riemannian symmetric spaces</a>.

The series without the factor of n! in the denominator (summed over all integers n, including negative) is called the <a href="Bilateral_hypergeometric_series" title="Bilateral hypergeometric series">bilateral hypergeometric series</a>.

<div class="mw-heading mw-heading2"><h2 id="Convergence_conditions">Convergence conditions</h2></div>

There are certain values of the aj and bk for which the numerator or the denominator of the coefficients is 0.

<ul><li>If any aj is a non-positive integer (0, −1, −2, etc.) then the series only has a finite number of terms and is, in fact, a polynomial of degree −aj.</li>

<li>If any bk is a non-positive integer (excepting the previous case with bk < aj) then the denominators become 0 and the series is undefined.</li></ul>

Excluding these cases, the <a href="Ratio_test" title="Ratio test">ratio test</a> can be applied to determine the radius of convergence.

<li>If p = q + 1 then the ratio of coefficients tends to one. This implies that the series converges for |z| < 1 and diverges for |z| > 1. Whether it converges for |z| = 1 is more difficult to determine. Analytic continuation can be employed for larger values of z.</li>

<li>If p > q + 1 then the ratio of coefficients grows without bound. This implies that, besides z = 0, the series diverges. This is then a divergent or asymptotic series, or it can be interpreted as a symbolic shorthand for a differential equation that the sum satisfies formally.</li></ul>

The question of convergence for p=q+1 when z is on the unit circle is more difficult. It can be shown that the series converges absolutely at z = 1 if

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<annotation encoding="application/x-tex">{\displaystyle \Re \left(\sum b_{k}-\sum a_{j}\right)>0}</annotation>

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</math><img src="./c620b54094a8c6c2da1b7140769ff390df233b84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.898ex; height:4.843ex;" alt="{\displaystyle \Re \left(\sum b_{k}-\sum a_{j}\right)>0}" loading="lazy">.</dd></dl>

Further, if p=q+1, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{p}a_{i}\geq \sum _{j=1}^{q}b_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{p}a_{i}\geq \sum _{j=1}^{q}b_{j}}</annotation>

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</math><img src="./3d23cfccd18e485c7bebab1ef52f7b6c13cf61e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.519ex; height:7.343ex;" alt="{\displaystyle \sum _{i=1}^{p}a_{i}\geq \sum _{j=1}^{q}b_{j}}" loading="lazy"> and z is real, then the following convergence result holds <a href="#CITEREFQuigleyWilsonWallsBedford2013">Quigley et al. (2013)</a>:

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<annotation encoding="application/x-tex">{\displaystyle \lim _{z\rightarrow 1}(1-z){\frac {d\log(_{p}F_{q}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};z^{p}))}{dz}}=\sum _{i=1}^{p}a_{i}-\sum _{j=1}^{q}b_{j}}</annotation>

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</math><img src="./bdc9b13da6f7e781b1f6d2495e111956bce4141b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:63.251ex; height:7.343ex;" alt="{\displaystyle \lim _{z\rightarrow 1}(1-z){\frac {d\log(_{p}F_{q}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};z^{p}))}{dz}}=\sum _{i=1}^{p}a_{i}-\sum _{j=1}^{q}b_{j}}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Basic_properties">Basic properties</h2></div>

It is immediate from the definition that the order of the parameters aj, or the order of the parameters bk can be changed without changing the value of the function. Also, if any of the parameters aj is equal to any of the parameters bk, then the matching parameters can be "cancelled out", with certain exceptions when the parameters are non-positive integers. For example,

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<annotation encoding="application/x-tex">{\displaystyle \,{}_{2}F_{1}(3,1;1;z)=\,{}_{2}F_{1}(1,3;1;z)=\,{}_{1}F_{0}(3;;z)}</annotation>

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</math><img src="./4d55b2d8e7cf91576ac42e13c31c2a5ff1895278.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.268ex; height:2.843ex;" alt="{\displaystyle \,{}_{2}F_{1}(3,1;1;z)=\,{}_{2}F_{1}(1,3;1;z)=\,{}_{1}F_{0}(3;;z)}" loading="lazy">.</dd></dl>

This cancelling is a special case of a reduction formula that may be applied whenever a parameter on the top row differs from one on the bottom row by a non-negative integer.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle {}_{A+1}F_{B+1}\left[{\begin{array}{c}a_{1},\ldots ,a_{A},c+n\\b_{1},\ldots ,b_{B},c\end{array}};z\right]=\sum _{j=0}^{n}{\binom {n}{j}}{\frac {z^{j}}{(c)_{j}}}{\frac {\prod _{i=1}^{A}(a_{i})_{j}}{\prod _{i=1}^{B}(b_{i})_{j}}}{}_{A}F_{B}\left[{\begin{array}{c}a_{1}+j,\ldots ,a_{A}+j\\b_{1}+j,\ldots ,b_{B}+j\end{array}};z\right]}</annotation>

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</math><img src="./52187cb597741e873af9b16d3df4f6f9a5df9321.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:85.594ex; height:7.843ex;" alt="{\displaystyle {}_{A+1}F_{B+1}\left[{\begin{array}{c}a_{1},\ldots ,a_{A},c+n\\b_{1},\ldots ,b_{B},c\end{array}};z\right]=\sum _{j=0}^{n}{\binom {n}{j}}{\frac {z^{j}}{(c)_{j}}}{\frac {\prod _{i=1}^{A}(a_{i})_{j}}{\prod _{i=1}^{B}(b_{i})_{j}}}{}_{A}F_{B}\left[{\begin{array}{c}a_{1}+j,\ldots ,a_{A}+j\\b_{1}+j,\ldots ,b_{B}+j\end{array}};z\right]}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Euler's_integral_transform">Euler's integral transform</h3></div>

The following basic identity is very useful as it relates the higher-order hypergeometric functions in terms of integrals over the lower order ones<a href="#cite_note-3">[3]</a>

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<annotation encoding="application/x-tex">{\displaystyle {}_{A+1}F_{B+1}\left[{\begin{array}{c}a_{1},\ldots ,a_{A},c\\b_{1},\ldots ,b_{B},d\end{array}};z\right]={\frac {\Gamma (d)}{\Gamma (c)\Gamma (d-c)}}\int _{0}^{1}t^{c-1}(1-t)_{}^{d-c-1}\ {}_{A}F_{B}\left[{\begin{array}{c}a_{1},\ldots ,a_{A}\\b_{1},\ldots ,b_{B}\end{array}};tz\right]dt}</annotation>

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</math><img src="./a2a5333d174c4b3dabe503bb002b2f686cf88f16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:87.268ex; height:6.509ex;" alt="{\displaystyle {}_{A+1}F_{B+1}\left[{\begin{array}{c}a_{1},\ldots ,a_{A},c\\b_{1},\ldots ,b_{B},d\end{array}};z\right]={\frac {\Gamma (d)}{\Gamma (c)\Gamma (d-c)}}\int _{0}^{1}t^{c-1}(1-t)_{}^{d-c-1}\ {}_{A}F_{B}\left[{\begin{array}{c}a_{1},\ldots ,a_{A}\\b_{1},\ldots ,b_{B}\end{array}};tz\right]dt}" loading="lazy"></dd></dl>

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

The generalized hypergeometric function satisfies

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+a_{j}\right){}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{j},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{array}};z\right]&=a_{j}\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{j}+1,\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{array}};z\right]\\\end{aligned}}}</annotation>

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</math><img src="./03f26c726fef7e97a364cc96b811cbc9ee2dbdd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:76.384ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+a_{j}\right){}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{j},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{array}};z\right]&=a_{j}\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{j}+1,\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{array}};z\right]\\\end{aligned}}}" loading="lazy"></dd></dl>

and

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+b_{k}-1\right){}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{k},\dots ,b_{q}\end{array}};z\right]&=(b_{k}-1)\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{k}-1,\dots ,b_{q}\end{array}};z\right]{\text{ for }}b_{k}\neq 1\end{aligned}}}</annotation>

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</math><img src="./a8ebc658f7845c47473d4f2f00ffdd7b6ef2001c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:95.598ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+b_{k}-1\right){}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{k},\dots ,b_{q}\end{array}};z\right]&=(b_{k}-1)\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{k}-1,\dots ,b_{q}\end{array}};z\right]{\text{ for }}b_{k}\neq 1\end{aligned}}}" loading="lazy">

Additionally,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\rm {d}}{{\rm {d}}z}}\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{array}};z\right]&={\frac {\prod _{i=1}^{p}a_{i}}{\prod _{j=1}^{q}b_{j}}}\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1}+1,\dots ,a_{p}+1\\b_{1}+1,\dots ,b_{q}+1\end{array}};z\right]\end{aligned}}}</annotation>

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</math><img src="./8591afd4f998294413c43b47a1c0a1bc874a11f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:62.728ex; height:7.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\rm {d}}{{\rm {d}}z}}\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{array}};z\right]&={\frac {\prod _{i=1}^{p}a_{i}}{\prod _{j=1}^{q}b_{j}}}\;{}_{p}F_{q}\left[{\begin{array}{c}a_{1}+1,\dots ,a_{p}+1\\b_{1}+1,\dots ,b_{q}+1\end{array}};z\right]\end{aligned}}}" loading="lazy">

Combining these gives a differential equation satisfied by w = pFq:

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<annotation encoding="application/x-tex">{\displaystyle z\prod _{n=1}^{p}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+a_{n}\right)w=z{\frac {\rm {d}}{{\rm {d}}z}}\prod _{n=1}^{q}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+b_{n}-1\right)w}</annotation>

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</math><img src="./46029e662abf93ee80b300bc9b1bb509841a98c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:50.116ex; height:7.009ex;" alt="{\displaystyle z\prod _{n=1}^{p}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+a_{n}\right)w=z{\frac {\rm {d}}{{\rm {d}}z}}\prod _{n=1}^{q}\left(z{\frac {\rm {d}}{{\rm {d}}z}}+b_{n}-1\right)w}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Contiguous_function_and_related_identities">Contiguous function and related identities</h2></div>

Take the following operator:

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<annotation encoding="application/x-tex">{\displaystyle \vartheta =z{\frac {\rm {d}}{{\rm {d}}z}}.}</annotation>

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</math><img src="./8da561258499de127b70194169f10c09b2b57b58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.424ex; height:5.509ex;" alt="{\displaystyle \vartheta =z{\frac {\rm {d}}{{\rm {d}}z}}.}" loading="lazy"></dd></dl>

From the differentiation formulas given above, the linear space spanned by

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<annotation encoding="application/x-tex">{\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z),\vartheta \;{}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z)}</annotation>

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</math><img src="./3f53a63e4d6d45ed99973fe5744b8e5f504cb564.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.063ex; width:58.066ex; height:3.009ex;" alt="{\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z),\vartheta \;{}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z)}" loading="lazy"></dd></dl>

contains each of

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<annotation encoding="application/x-tex">{\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{j}+1,\dots ,a_{p};b_{1},\dots ,b_{q};z),}</annotation>

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</math><img src="./18750f3c8bcfcf31fe577268136d484d09de1cff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.063ex; width:39.506ex; height:3.009ex;" alt="{\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{j}+1,\dots ,a_{p};b_{1},\dots ,b_{q};z),}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{k}-1,\dots ,b_{q};z),}</annotation>

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</math><img src="./621496fbd069ac433dc0f78f0d8feca8dfb0f36b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.063ex; width:39.452ex; height:3.009ex;" alt="{\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{k}-1,\dots ,b_{q};z),}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle z\;{}_{p}F_{q}(a_{1}+1,\dots ,a_{p}+1;b_{1}+1,\dots ,b_{q}+1;z),}</annotation>

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</math><img src="./e3873dc1b3d6e50a6ec7066940be0cf76e199a9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:45.866ex; height:3.009ex;" alt="{\displaystyle z\;{}_{p}F_{q}(a_{1}+1,\dots ,a_{p}+1;b_{1}+1,\dots ,b_{q}+1;z),}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z).}</annotation>

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</math><img src="./c921a1a18dfdf3e64afe1fccc1da2b589e33f8e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.063ex; width:28.185ex; height:3.009ex;" alt="{\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z).}" loading="lazy"></dd></dl>

Since the space has dimension 2, any three of these p+q+2 functions are linearly dependent:

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<annotation encoding="application/x-tex">{\displaystyle (a_{i}-b_{j}+1){}_{p}F_{q}(...a_{i}..;...,b_{j}...;z)=a_{i}\,{}_{p}F_{q}(...a_{i}+1..;...,b_{j}...;z)-(b_{j}-1){}_{p}F_{q}(...a_{i}..;...,b_{j}-1...;z).}</annotation>

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</math><img src="./8b3014b3fb188240eebd311b5128aeb396d5a3b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:109.936ex; height:3.009ex;" alt="{\displaystyle (a_{i}-b_{j}+1){}_{p}F_{q}(...a_{i}..;...,b_{j}...;z)=a_{i}\,{}_{p}F_{q}(...a_{i}+1..;...,b_{j}...;z)-(b_{j}-1){}_{p}F_{q}(...a_{i}..;...,b_{j}-1...;z).}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle (a_{i}-a_{j}){}_{p}F_{q}(...a_{i}..a_{j}..;.....;z)=a_{i}\,{}_{p}F_{q}(...a_{i}+1..a_{j}..;......;z)-a_{j}\,{}_{p}F_{q}(...a_{i}..a_{j}+1...;....;z).}</annotation>

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</math><img src="./059c1f5062dc5945dc91916ab0b72b8dc1adb547.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:102.703ex; height:3.009ex;" alt="{\displaystyle (a_{i}-a_{j}){}_{p}F_{q}(...a_{i}..a_{j}..;.....;z)=a_{i}\,{}_{p}F_{q}(...a_{i}+1..a_{j}..;......;z)-a_{j}\,{}_{p}F_{q}(...a_{i}..a_{j}+1...;....;z).}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle b_{j}\,{}_{p}F_{q}(...a_{i}....;..b_{j}...;z)=a_{i}\,{}_{p}F_{q}(...a_{i}+1....;..b_{j}+1...;z)+(b_{j}-a_{i}){}_{p}F_{q}(...a_{i}....;..b_{j}+1...;z).}</annotation>

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</math><img src="./5c13e714a49e1b6990b78c6e31c0eabf87bc2e71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:102.576ex; height:3.009ex;" alt="{\displaystyle b_{j}\,{}_{p}F_{q}(...a_{i}....;..b_{j}...;z)=a_{i}\,{}_{p}F_{q}(...a_{i}+1....;..b_{j}+1...;z)+(b_{j}-a_{i}){}_{p}F_{q}(...a_{i}....;..b_{j}+1...;z).}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle (a_{i}-1){}_{p}F_{q}(...a_{i}..a_{j};...;z)=(a_{i}-a_{j}-1){}_{p}F_{q}(...a_{i}-1..a_{j};...;z)+a_{j}\,{}_{p}F_{q}(...a_{i}-1..a_{j}+1;...;z).}</annotation>

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</math><img src="./3467b34faac67c1c22532f90636771da4759a7c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:103.08ex; height:3.009ex;" alt="{\displaystyle (a_{i}-1){}_{p}F_{q}(...a_{i}..a_{j};...;z)=(a_{i}-a_{j}-1){}_{p}F_{q}(...a_{i}-1..a_{j};...;z)+a_{j}\,{}_{p}F_{q}(...a_{i}-1..a_{j}+1;...;z).}" loading="lazy"></dd></dl>

These dependencies can be written out to generate a large number of identities involving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{p}F_{q}}">

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

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</math><img src="./7fc3dfcd727e24842241f2a4f195e830213e4e6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.063ex; width:3.606ex; height:2.843ex;" alt="{\displaystyle {}_{p}F_{q}}" loading="lazy">.

For example, in the simplest non-trivial case,

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<annotation encoding="application/x-tex">{\displaystyle \;{}_{0}F_{1}(;a;z)=(1)\;{}_{0}F_{1}(;a;z)}</annotation>

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</math><img src="./4bd684d30f5a8c73212275fbd53f2d805e7fe28f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.957ex; height:2.843ex;" alt="{\displaystyle \;{}_{0}F_{1}(;a;z)=(1)\;{}_{0}F_{1}(;a;z)}" loading="lazy">,</dd>

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<annotation encoding="application/x-tex">{\displaystyle \;{}_{0}F_{1}(;a-1;z)=({\frac {\vartheta }{a-1}}+1)\;{}_{0}F_{1}(;a;z)}</annotation>

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</math><img src="./f804629f2ff689a129f7efc93b7849456f481ba2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:39.869ex; height:5.509ex;" alt="{\displaystyle \;{}_{0}F_{1}(;a-1;z)=({\frac {\vartheta }{a-1}}+1)\;{}_{0}F_{1}(;a;z)}" loading="lazy">,</dd>

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<annotation encoding="application/x-tex">{\displaystyle z\;{}_{0}F_{1}(;a+1;z)=(a\vartheta )\;{}_{0}F_{1}(;a;z)}</annotation>

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</math><img src="./61a3c86f2e8a9f269791d604ba2247c2a2dcc77b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.489ex; height:2.843ex;" alt="{\displaystyle z\;{}_{0}F_{1}(;a+1;z)=(a\vartheta )\;{}_{0}F_{1}(;a;z)}" loading="lazy">,</dd></dl>

So

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<annotation encoding="application/x-tex">{\displaystyle \;{}_{0}F_{1}(;a-1;z)-\;{}_{0}F_{1}(;a;z)={\frac {z}{a(a-1)}}\;{}_{0}F_{1}(;a+1;z)}</annotation>

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</math><img src="./4a0cf1b46eeaeb3058edf49bc8b4b5a02b1f7dc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:54.383ex; height:5.509ex;" alt="{\displaystyle \;{}_{0}F_{1}(;a-1;z)-\;{}_{0}F_{1}(;a;z)={\frac {z}{a(a-1)}}\;{}_{0}F_{1}(;a+1;z)}" loading="lazy">.</dd></dl>

This, and other important examples,

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<annotation encoding="application/x-tex">{\displaystyle \;{}_{1}F_{1}(a+1;b;z)-\,{}_{1}F_{1}(a;b;z)={\frac {z}{b}}\;{}_{1}F_{1}(a+1;b+1;z)}</annotation>

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</math><img src="./8d9b2b32537cf4661c70b9aa1e1f471f55dbfe38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:53.936ex; height:4.843ex;" alt="{\displaystyle \;{}_{1}F_{1}(a+1;b;z)-\,{}_{1}F_{1}(a;b;z)={\frac {z}{b}}\;{}_{1}F_{1}(a+1;b+1;z)}" loading="lazy">,</dd>

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<annotation encoding="application/x-tex">{\displaystyle \;{}_{1}F_{1}(a;b-1;z)-\,{}_{1}F_{1}(a;b;z)={\frac {az}{b(b-1)}}\;{}_{1}F_{1}(a+1;b+1;z)}</annotation>

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</math><img src="./866781322be035ae2d04b1cd0d573956c1cfe106.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.655ex; height:5.509ex;" alt="{\displaystyle \;{}_{1}F_{1}(a;b-1;z)-\,{}_{1}F_{1}(a;b;z)={\frac {az}{b(b-1)}}\;{}_{1}F_{1}(a+1;b+1;z)}" loading="lazy">,</dd>

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<annotation encoding="application/x-tex">{\displaystyle \;{}_{1}F_{1}(a;b-1;z)-\,{}_{1}F_{1}(a+1;b;z)={\frac {(a-b+1)z}{b(b-1)}}\;{}_{1}F_{1}(a+1;b+1;z)}</annotation>

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</math><img src="./16b86bb45bd9711e0c32ccdc3a4ea2122c9f76a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:68.819ex; height:6.509ex;" alt="{\displaystyle \;{}_{1}F_{1}(a;b-1;z)-\,{}_{1}F_{1}(a+1;b;z)={\frac {(a-b+1)z}{b(b-1)}}\;{}_{1}F_{1}(a+1;b+1;z)}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle \;{}_{2}F_{1}(a+1,b;c;z)-\,{}_{2}F_{1}(a,b;c;z)={\frac {bz}{c}}\;{}_{2}F_{1}(a+1,b+1;c+1;z)}</annotation>

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</math><img src="./8a38af82a14e337219257025d243d88b26437e7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:65.059ex; height:5.343ex;" alt="{\displaystyle \;{}_{2}F_{1}(a+1,b;c;z)-\,{}_{2}F_{1}(a,b;c;z)={\frac {bz}{c}}\;{}_{2}F_{1}(a+1,b+1;c+1;z)}" loading="lazy">,</dd>

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<annotation encoding="application/x-tex">{\displaystyle \;{}_{2}F_{1}(a+1,b;c;z)-\,{}_{2}F_{1}(a,b+1;c;z)={\frac {(b-a)z}{c}}\;{}_{2}F_{1}(a+1,b+1;c+1;z)}</annotation>

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</math><img src="./97f3efb8b9f23c44a427e35c2e944c2b2289872f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:74.941ex; height:5.676ex;" alt="{\displaystyle \;{}_{2}F_{1}(a+1,b;c;z)-\,{}_{2}F_{1}(a,b+1;c;z)={\frac {(b-a)z}{c}}\;{}_{2}F_{1}(a+1,b+1;c+1;z)}" loading="lazy">,</dd>

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<annotation encoding="application/x-tex">{\displaystyle \;{}_{2}F_{1}(a,b;c-1;z)-\,{}_{2}F_{1}(a+1,b;c;z)={\frac {(a-c+1)bz}{c(c-1)}}\;{}_{2}F_{1}(a+1,b+1;c+1;z)}</annotation>

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</math><img src="./0a24924794507eae52d670f8b935fee9a364e67c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:79.951ex; height:6.509ex;" alt="{\displaystyle \;{}_{2}F_{1}(a,b;c-1;z)-\,{}_{2}F_{1}(a+1,b;c;z)={\frac {(a-c+1)bz}{c(c-1)}}\;{}_{2}F_{1}(a+1,b+1;c+1;z)}" loading="lazy">,</dd></dl>

can be used to generate <a href="Continued_fraction" title="Continued fraction">continued fraction</a> expressions known as <a href="Gauss's_continued_fraction" title="Gauss's continued fraction">Gauss's continued fraction</a>.

Similarly, by applying the differentiation formulas twice, there are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\binom {p+q+3}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\binom {p+q+3}{2}}}</annotation>

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</math><img src="./4f6d58d51158770f256b4aeaabfd9886172a18b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.503ex; height:6.176ex;" alt="{\displaystyle {\binom {p+q+3}{2}}}" loading="lazy"> such functions contained in

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<annotation encoding="application/x-tex">{\displaystyle \{1,\vartheta ,\vartheta ^{2}\}\;{}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z),}</annotation>

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</math><img src="./0b7f882fcffb60973d06bf88ac7f73a69f865883.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.124ex; height:3.343ex;" alt="{\displaystyle \{1,\vartheta ,\vartheta ^{2}\}\;{}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z),}" loading="lazy"></dd></dl>

which has dimension three so any four are linearly dependent. This generates more identities and the process can be continued. The identities thus generated can be combined with each other to produce new ones in a different way.

A function obtained by adding ±1 to exactly one of the parameters aj, bk in

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<annotation encoding="application/x-tex">{\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z)}</annotation>

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</math><img src="./23840450e3427c4643b74107257d6ef2448e7c51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.063ex; width:27.538ex; height:3.009ex;" alt="{\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z)}" loading="lazy"></dd></dl>

is called contiguous to

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<annotation encoding="application/x-tex">{\displaystyle {}_{p}F_{q}(a_{1},\dots ,a_{p};b_{1},\dots ,b_{q};z).}</annotation>

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Using the technique outlined above, an identity relating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{0}F_{1}(;a;z)}">

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<annotation encoding="application/x-tex">{\displaystyle {}_{1}F_{1}(a;b;z)}</annotation>

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</math><img src="./9f6d175be5530757c1e9f13e481ead1918942fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.796ex; height:2.843ex;" alt="{\displaystyle {}_{1}F_{1}(a;b;z)}" loading="lazy"> and any two of its four contiguous functions, and fifteen identities relating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{2}F_{1}(a,b;c;z)}">

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<annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}(a,b;c;z)}</annotation>

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

<div role="note" class="hatnote navigation-not-searchable">For identities involving the Gauss hypergeometric function 2F1, see <a href="Hypergeometric_function" title="Hypergeometric function">Hypergeometric function</a>.</div>

A number of other hypergeometric function identities were discovered in the nineteenth and twentieth centuries. A 20th century contribution to the methodology of proving these identities is the <a href="Egorychev_method" title="Egorychev method">Egorychev method</a>.

<div class="mw-heading mw-heading3"><h3 id="Saalschütz's_theorem">Saalschütz's theorem</h3></div>

Saalschütz's theorem<a href="#cite_note-6">[6]</a> (<a href="#CITEREFSaalschütz1890">Saalschütz 1890</a>) is

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<annotation encoding="application/x-tex">{\displaystyle {}_{3}F_{2}(a,b,-n;c,1+a+b-c-n;1)={\frac {(c-a)_{n}(c-b)_{n}}{(c)_{n}(c-a-b)_{n}}}.}</annotation>

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</math><img src="./737370c34bde92dc5ecd289d356f02c1b9ff5033.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:55.893ex; height:6.509ex;" alt="{\displaystyle {}_{3}F_{2}(a,b,-n;c,1+a+b-c-n;1)={\frac {(c-a)_{n}(c-b)_{n}}{(c)_{n}(c-a-b)_{n}}}.}" loading="lazy"></dd></dl>

For extension of this theorem, see a research paper by Rakha & Rathie. According to (<a href="#CITEREFAndrewsAskeyRoy1999">Andrews, Askey & Roy 1999</a>, p. 69), it was in fact first discovered by <a href="Johann_Friedrich_Pfaff" title="Johann Friedrich Pfaff">Pfaff</a> in 1797.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading3"><h3 id="Dixon's_identity">Dixon's identity</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Dixon's_identity" title="Dixon's identity">Dixon's identity</a></div>

Dixon's identity,<a href="#cite_note-8">[8]</a> first proved by <a href="#CITEREFDixon1902">Dixon (1902)</a>, gives the sum of a well-poised 3F2 at 1:

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<annotation encoding="application/x-tex">{\displaystyle {}_{3}F_{2}(a,b,c;1+a-b,1+a-c;1)={\frac {\Gamma (1+{\frac {a}{2}})\Gamma (1+{\frac {a}{2}}-b-c)\Gamma (1+a-b)\Gamma (1+a-c)}{\Gamma (1+a)\Gamma (1+a-b-c)\Gamma (1+{\frac {a}{2}}-b)\Gamma (1+{\frac {a}{2}}-c)}}.}</annotation>

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</math><img src="./864a5eff01238a9dc94ebbf783d814959bb876ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:88.012ex; height:7.176ex;" alt="{\displaystyle {}_{3}F_{2}(a,b,c;1+a-b,1+a-c;1)={\frac {\Gamma (1+{\frac {a}{2}})\Gamma (1+{\frac {a}{2}}-b-c)\Gamma (1+a-b)\Gamma (1+a-c)}{\Gamma (1+a)\Gamma (1+a-b-c)\Gamma (1+{\frac {a}{2}}-b)\Gamma (1+{\frac {a}{2}}-c)}}.}" loading="lazy"></dd></dl>

For generalization of Dixon's identity, see a paper by Lavoie, et al.

<div class="mw-heading mw-heading3"><h3 id="Dougall's_formula">Dougall's formula</h3></div>

Dougall's formula (<a href="John_Dougall_(mathematician)" title="John Dougall (mathematician)">Dougall</a> <a href="#CITEREFDougall1907">1907</a>) gives the sum of a very well-poised series that is terminating and 2-balanced.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{}_{7}F_{6}&\left({\begin{matrix}a&1+{\frac {a}{2}}&b&c&d&e&-m\\&{\frac {a}{2}}&1+a-b&1+a-c&1+a-d&1+a-e&1+a+m\\\end{matrix}};1\right)=\\&={\frac {(1+a)_{m}(1+a-b-c)_{m}(1+a-c-d)_{m}(1+a-b-d)_{m}}{(1+a-b)_{m}(1+a-c)_{m}(1+a-d)_{m}(1+a-b-c-d)_{m}}}.\end{aligned}}}</annotation>

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</math><img src="./2eeafc61e7cc81b0b65d769ee71df13f4034ec92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:81.02ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}{}_{7}F_{6}&\left({\begin{matrix}a&1+{\frac {a}{2}}&b&c&d&e&-m\\&{\frac {a}{2}}&1+a-b&1+a-c&1+a-d&1+a-e&1+a+m\\\end{matrix}};1\right)=\\&={\frac {(1+a)_{m}(1+a-b-c)_{m}(1+a-c-d)_{m}(1+a-b-d)_{m}}{(1+a-b)_{m}(1+a-c)_{m}(1+a-d)_{m}(1+a-b-c-d)_{m}}}.\end{aligned}}}" loading="lazy"></dd></dl>

Terminating means that m is a non-negative integer and 2-balanced means that

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<annotation encoding="application/x-tex">{\displaystyle 1+2a=b+c+d+e-m.}</annotation>

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</math><img src="./a8d3f3cf73a3ccd690a75a4e4a7ad3522766cc80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:27.846ex; height:2.343ex;" alt="{\displaystyle 1+2a=b+c+d+e-m.}" loading="lazy"></dd></dl>

Many of the other formulas for special values of hypergeometric functions can be derived from this as special or limiting cases. It is also called the Dougall-Ramanujan identity. It is a special case of Jackson's identity, and it gives Dixon's identity and Saalschütz's theorem as special cases.<a href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Generalization_of_Kummer's_transformations_and_identities_for_2F2">Generalization of Kummer's transformations and identities for 2F2</h3></div>

Identity 1.

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<annotation encoding="application/x-tex">{\displaystyle e^{-x}\;{}_{2}F_{2}(a,1+d;c,d;x)={}_{2}F_{2}(c-a-1,f+1;c,f;-x)}</annotation>

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</math><img src="./ae6e8346bf2a7da96f9f34bdf95543331b2e5742.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.16ex; height:3.009ex;" alt="{\displaystyle e^{-x}\;{}_{2}F_{2}(a,1+d;c,d;x)={}_{2}F_{2}(c-a-1,f+1;c,f;-x)}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle f={\frac {d(a-c+1)}{a-d}}}</annotation>

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</math><img src="./688f560dd5ff3cf8c98501d0cb44365745fa70f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.318ex; height:6.009ex;" alt="{\displaystyle f={\frac {d(a-c+1)}{a-d}}}" loading="lazy">;</dd></dl>

Identity 2.

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<annotation encoding="application/x-tex">{\displaystyle e^{-{\frac {x}{2}}}\,{}_{2}F_{2}\left(a,1+b;2a+1,b;x\right)={}_{0}F_{1}\left(;a+{\tfrac {1}{2}};{\tfrac {x^{2}}{16}}\right)-{\frac {x\left(1-{\tfrac {2a}{b}}\right)}{2(2a+1)}}\;{}_{0}F_{1}\left(;a+{\tfrac {3}{2}};{\tfrac {x^{2}}{16}}\right),}</annotation>

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</math><img src="./56b37dbc3225cbac9d8b0277e59a9c3d999df211.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:83.285ex; height:8.343ex;" alt="{\displaystyle e^{-{\frac {x}{2}}}\,{}_{2}F_{2}\left(a,1+b;2a+1,b;x\right)={}_{0}F_{1}\left(;a+{\tfrac {1}{2}};{\tfrac {x^{2}}{16}}\right)-{\frac {x\left(1-{\tfrac {2a}{b}}\right)}{2(2a+1)}}\;{}_{0}F_{1}\left(;a+{\tfrac {3}{2}};{\tfrac {x^{2}}{16}}\right),}" loading="lazy"></dd></dl>

which links <a href="Bessel_function" title="Bessel function">Bessel functions</a> to 2F2; this reduces to Kummer's second formula for b = 2a:

Identity 3.

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<annotation encoding="application/x-tex">{\displaystyle e^{-{\frac {x}{2}}}\,{}_{1}F_{1}(a,2a,x)={}_{0}F_{1}\left(;a+{\tfrac {1}{2}};{\tfrac {x^{2}}{16}}\right)}</annotation>

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</math><img src="./0b1b504af5b0725090d984fca7c4dae4af4196ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.281ex; height:4.843ex;" alt="{\displaystyle e^{-{\frac {x}{2}}}\,{}_{1}F_{1}(a,2a,x)={}_{0}F_{1}\left(;a+{\tfrac {1}{2}};{\tfrac {x^{2}}{16}}\right)}" loading="lazy">.</dd></dl>

Identity 4.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{}_{2}F_{2}(a,b;c,d;x)=&\sum _{i=0}{\frac {{b-d \choose i}{a+i-1 \choose i}}{{c+i-1 \choose i}{d+i-1 \choose i}}}\;{}_{1}F_{1}(a+i;c+i;x){\frac {x^{i}}{i!}}\\=&e^{x}\sum _{i=0}{\frac {{b-d \choose i}{a+i-1 \choose i}}{{c+i-1 \choose i}{d+i-1 \choose i}}}\;{}_{1}F_{1}(c-a;c+i;-x){\frac {x^{i}}{i!}},\end{aligned}}}</annotation>

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</math><img src="./260cf839970b88d4a7b5bc335f3394ed8c146ded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:64.081ex; height:15.676ex;" alt="{\displaystyle {\begin{aligned}{}_{2}F_{2}(a,b;c,d;x)=&\sum _{i=0}{\frac {{b-d \choose i}{a+i-1 \choose i}}{{c+i-1 \choose i}{d+i-1 \choose i}}}\;{}_{1}F_{1}(a+i;c+i;x){\frac {x^{i}}{i!}}\\=&e^{x}\sum _{i=0}{\frac {{b-d \choose i}{a+i-1 \choose i}}{{c+i-1 \choose i}{d+i-1 \choose i}}}\;{}_{1}F_{1}(c-a;c+i;-x){\frac {x^{i}}{i!}},\end{aligned}}}" loading="lazy"></dd></dl>

which is a finite sum if b-d is a non-negative integer.

<div class="mw-heading mw-heading3"><h3 id="Kummer's_relation">Kummer's relation</h3></div>

Kummer's relation is

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}\left(2a,2b;a+b+{\tfrac {1}{2}};x\right)={}_{2}F_{1}\left(a,b;a+b+{\tfrac {1}{2}};4x(1-x)\right).}</annotation>

</semantics>

</math><img src="./3e72911cfa388d0c10654dbab3a34721df525ff0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:59.452ex; height:3.509ex;" alt="{\displaystyle {}_{2}F_{1}\left(2a,2b;a+b+{\tfrac {1}{2}};x\right)={}_{2}F_{1}\left(a,b;a+b+{\tfrac {1}{2}};4x(1-x)\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Clausen's_formula">Clausen's formula</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Clausen's_formula" title="Clausen's formula">Clausen's formula</a></div>

Clausen's formula

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{3}F_{2}(2c-2s-1,2s,c-{\tfrac {1}{2}};2c-1,c;x)=\,{}_{2}F_{1}(c-s-{\tfrac {1}{2}},s;c;x)^{2}}</annotation>

</semantics>

</math><img src="./ee9dfb73f10ceefd803d551c9ba3908159caa432.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:64.032ex; height:3.509ex;" alt="{\displaystyle {}_{3}F_{2}(2c-2s-1,2s,c-{\tfrac {1}{2}};2c-1,c;x)=\,{}_{2}F_{1}(c-s-{\tfrac {1}{2}},s;c;x)^{2}}" loading="lazy"></dd></dl>

was used by <a href="Louis_de_Branges_de_Bourcia" title="Louis de Branges de Bourcia">de Branges</a> to prove the <a href="Bieberbach_conjecture" class="mw-redirect" title="Bieberbach conjecture">Bieberbach conjecture</a>.

<div class="mw-heading mw-heading2"><h2 id="Special_cases">Special cases</h2></div>

Many of the special functions in mathematics are special cases of the <a href="Confluent_hypergeometric_function" title="Confluent hypergeometric function">confluent hypergeometric function</a> or the <a href="Hypergeometric_function" title="Hypergeometric function">hypergeometric function</a>; see the corresponding articles for examples.

<div class="mw-heading mw-heading3"><h3 id="The_series_0F0">The series 0F0</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Exponential_function" title="Exponential function">Exponential function</a></div>

As noted earlier, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{0}F_{0}(;;z)=e^{z}}">

<msub>

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</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{0}F_{0}(;;z)=e^{z}}</annotation>

</semantics>

</math><img src="./f7240ab48b322df73458156a81950b31f12f7a57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.752ex; height:2.843ex;" alt="{\displaystyle {}_{0}F_{0}(;;z)=e^{z}}" loading="lazy">. The differential equation for this function is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dz}}w=w}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dz}}w=w}</annotation>

</semantics>

</math><img src="./51bdad3104c1c8bb1882b0dbe9bb687b274ba64d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.567ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dz}}w=w}" loading="lazy">, which has solutions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=ke^{z}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./142e1df300ce18c5e4e056c472110269946abaac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.059ex; height:2.343ex;" alt="{\displaystyle w=ke^{z}}" loading="lazy"> where k is a constant.

<div class="mw-heading mw-heading3"><h3 id="The_series_0F1">The series 0F1</h3></div>

The functions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{0}F_{1}(;a;z)}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{0}F_{1}(;a;z)}</annotation>

</semantics>

</math><img src="./f5f137cbf6624a12fa46a7203246cebecd4e3df3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.798ex; height:2.843ex;" alt="{\displaystyle {}_{0}F_{1}(;a;z)}" loading="lazy"> are called confluent hypergeometric limit functions and are closely related to <a href="Bessel_function" title="Bessel function">Bessel functions</a>.

The relationship is:

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{\alpha }(x)={\frac {({\tfrac {x}{2}})^{\alpha }}{\Gamma (\alpha +1)}}{}_{0}F_{1}\left(;\alpha +1;-{\tfrac {1}{4}}x^{2}\right).}</annotation>

</semantics>

</math><img src="./ad8d129ed604c91b3ccc71f6044659f24c5a6abf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.609ex; height:6.843ex;" alt="{\displaystyle J_{\alpha }(x)={\frac {({\tfrac {x}{2}})^{\alpha }}{\Gamma (\alpha +1)}}{}_{0}F_{1}\left(;\alpha +1;-{\tfrac {1}{4}}x^{2}\right).}" loading="lazy"></dd>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{\alpha }(x)={\frac {({\tfrac {x}{2}})^{\alpha }}{\Gamma (\alpha +1)}}{}_{0}F_{1}\left(;\alpha +1;{\tfrac {1}{4}}x^{2}\right).}</annotation>

</semantics>

</math><img src="./24fa2ea82e617cf429e453a2b63ea0e6b62d77de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.533ex; height:6.843ex;" alt="{\displaystyle I_{\alpha }(x)={\frac {({\tfrac {x}{2}})^{\alpha }}{\Gamma (\alpha +1)}}{}_{0}F_{1}\left(;\alpha +1;{\tfrac {1}{4}}x^{2}\right).}" loading="lazy"></dd></dl>

The differential equation for this function is

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w=\left(z{\frac {d}{dz}}+a\right){\frac {dw}{dz}}}</annotation>

</semantics>

</math><img src="./db552f0ecf33866c0b519b397100731b5237cf24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.585ex; height:6.176ex;" alt="{\displaystyle w=\left(z{\frac {d}{dz}}+a\right){\frac {dw}{dz}}}" loading="lazy"></dd></dl>

or

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+a{\frac {dw}{dz}}-w=0.}</annotation>

</semantics>

</math><img src="./294dff2f2bdfa2222284e2c6778b4f0a2c077f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.059ex; height:6.009ex;" alt="{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+a{\frac {dw}{dz}}-w=0.}" loading="lazy"></dd></dl>

When a is not a positive integer, the substitution

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w=z^{1-a}u,}</annotation>

</semantics>

</math><img src="./893a131e096ded74f8be6de52b934b846e6d4037.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.032ex; height:3.009ex;" alt="{\displaystyle w=z^{1-a}u,}" loading="lazy"></dd></dl>

gives a linearly independent solution

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z^{1-a}\;{}_{0}F_{1}(;2-a;z),}</annotation>

</semantics>

</math><img src="./6bf0b466865666a73495411a576aace8c464c9ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.386ex; height:3.176ex;" alt="{\displaystyle z^{1-a}\;{}_{0}F_{1}(;2-a;z),}" loading="lazy"></dd></dl>

so the general solution is

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\;{}_{0}F_{1}(;a;z)+lz^{1-a}\;{}_{0}F_{1}(;2-a;z)}</annotation>

</semantics>

</math><img src="./090f8330c5ecfdae2b841429fb15a1889e465478.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.927ex; height:3.176ex;" alt="{\displaystyle k\;{}_{0}F_{1}(;a;z)+lz^{1-a}\;{}_{0}F_{1}(;2-a;z)}" loading="lazy"></dd></dl>

where k, l are constants. (If a is a positive integer, the independent solution is given by the appropriate Bessel function of the second kind.)

A special case is:

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{0}F_{1}\left(;{\frac {1}{2}};-{\frac {z^{2}}{4}}\right)=\cos z}</annotation>

</semantics>

</math><img src="./c4105d986be746b8bf5885859029444730b795bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.951ex; height:6.343ex;" alt="{\displaystyle {}_{0}F_{1}\left(;{\frac {1}{2}};-{\frac {z^{2}}{4}}\right)=\cos z}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="The_series_1F0">The series 1F0</h3></div>

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

An important case is:

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{1}F_{0}(a;;z)=(1-z)^{-a}.}</annotation>

</semantics>

</math><img src="./02835bd6054a3a63e1e6672b62bcb86aa3e62a5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.824ex; height:3.009ex;" alt="{\displaystyle {}_{1}F_{0}(a;;z)=(1-z)^{-a}.}" loading="lazy"></dd></dl>

The differential equation for this function is

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dz}}w=\left(z{\frac {d}{dz}}+a\right)w,}</annotation>

</semantics>

</math><img src="./e53158558cdf1088f3aa2541742028eef0617c2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.32ex; height:6.176ex;" alt="{\displaystyle {\frac {d}{dz}}w=\left(z{\frac {d}{dz}}+a\right)w,}" loading="lazy"></dd></dl>

or

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./72891df0a56b7cfd53f1ee4fb76ad74e029cb8f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.256ex; height:5.509ex;" alt="{\displaystyle (1-z){\frac {dw}{dz}}=aw,}" loading="lazy"></dd></dl>

which has solutions

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./03edba0c3b17d55ccb6e9dbe81ba8e4aeb42b36c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.255ex; height:3.009ex;" alt="{\displaystyle w=k(1-z)^{-a}}" loading="lazy"></dd></dl>

where k is a constant.

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{1}F_{0}(1;;z)=\sum _{n\geqslant 0}z^{n}=(1-z)^{-1}}</annotation>

</semantics>

</math><img src="./de779f5f548bd72b69994b67600dbe804f89c18e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.212ex; height:5.676ex;" alt="{\displaystyle {}_{1}F_{0}(1;;z)=\sum _{n\geqslant 0}z^{n}=(1-z)^{-1}}" loading="lazy"> is the <a href="Geometric_series" title="Geometric series">geometric series</a> with ratio z and coefficient 1.</dd></dl>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z~{}_{1}F_{0}(2;;z)=\sum _{n\geqslant 0}nz^{n}=z(1-z)^{-2}}</annotation>

</semantics>

</math><img src="./e8de8d699323a22d0125360c6e7f5cb4dcf49842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.363ex; height:5.676ex;" alt="{\displaystyle z~{}_{1}F_{0}(2;;z)=\sum _{n\geqslant 0}nz^{n}=z(1-z)^{-2}}" loading="lazy"> is also useful.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="The_series_1F1">The series 1F1</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Confluent_hypergeometric_function" title="Confluent hypergeometric function">Confluent hypergeometric function</a></div>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {}_{1}F_{1}(a;b;z)}</annotation>

</semantics>

</math><img src="./9f6d175be5530757c1e9f13e481ead1918942fe6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.796ex; height:2.843ex;" alt="{\displaystyle {}_{1}F_{1}(a;b;z)}" loading="lazy"> are called confluent hypergeometric functions of the first kind, also written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(a;b;z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(a;b;z)}</annotation>

</semantics>

</math><img src="./8708ddf3194285a826de019c32b9201cf6db2e0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.635ex; height:2.843ex;" alt="{\displaystyle M(a;b;z)}" loading="lazy">. The incomplete gamma function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (a,z)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./fce6a934dffb9f82967b11bbbbb6f65b5180e17c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.424ex; height:2.843ex;" alt="{\displaystyle \gamma (a,z)}" loading="lazy"> is a special case.

The differential equation for this function is

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(z{\frac {d}{dz}}+a\right)w=\left(z{\frac {d}{dz}}+b\right){\frac {dw}{dz}}}</annotation>

</semantics>

</math><img src="./95569431d02c57077e6e8f6c77c89c7cf8406231.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.46ex; height:6.176ex;" alt="{\displaystyle \left(z{\frac {d}{dz}}+a\right)w=\left(z{\frac {d}{dz}}+b\right){\frac {dw}{dz}}}" loading="lazy"></dd></dl>

or

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0.}</annotation>

</semantics>

</math><img src="./cb1b3bd0c88cbeb74392c5a73606799ae609bacf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.795ex; height:6.009ex;" alt="{\displaystyle z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0.}" loading="lazy"></dd></dl>

When b is not a positive integer, the substitution

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<annotation encoding="application/x-tex">{\displaystyle w=z^{1-b}u,}</annotation>

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</math><img src="./0c1370c0bb2f4d634660daf48f272b44822cec12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.868ex; height:3.009ex;" alt="{\displaystyle w=z^{1-b}u,}" loading="lazy"></dd></dl>

gives a linearly independent solution

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<annotation encoding="application/x-tex">{\displaystyle z^{1-b}\;{}_{1}F_{1}(1+a-b;2-b;z),}</annotation>

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</math><img src="./8715886a351f9b179c4122baf7a4f6515a910054.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.06ex; height:3.176ex;" alt="{\displaystyle z^{1-b}\;{}_{1}F_{1}(1+a-b;2-b;z),}" loading="lazy"></dd></dl>

so the general solution is

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<annotation encoding="application/x-tex">{\displaystyle k\;{}_{1}F_{1}(a;b;z)+lz^{1-b}\;{}_{1}F_{1}(1+a-b;2-b;z)}</annotation>

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</math><img src="./2be12ccdc87835bf0da92760f48534d6d10df1db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.599ex; height:3.176ex;" alt="{\displaystyle k\;{}_{1}F_{1}(a;b;z)+lz^{1-b}\;{}_{1}F_{1}(1+a-b;2-b;z)}" loading="lazy"></dd></dl>

where k, l are constants.

When a is a non-positive integer, −n, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{1}F_{1}(-n;b;z)}">

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<annotation encoding="application/x-tex">{\displaystyle {}_{1}F_{1}(-n;b;z)}</annotation>

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</math><img src="./b3bb80b54c106698adf5b5a7b89ae062dd99f3c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.769ex; height:2.843ex;" alt="{\displaystyle {}_{1}F_{1}(-n;b;z)}" loading="lazy"> is a polynomial. Up to constant factors, these are the <a href="Laguerre_polynomials" title="Laguerre polynomials">Laguerre polynomials</a>. This implies <a href="Hermite_polynomials" title="Hermite polynomials">Hermite polynomials</a> can be expressed in terms of 1F1 as well.

<div class="mw-heading mw-heading3"><h3 id="The_series_1F2">The series 1F2</h3></div>

Relations to other functions are known for certain parameter combinations only.

The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\;{}_{1}F_{2}\left({\frac {1}{2}};{\frac {3}{2}},{\frac {3}{2}};-{\frac {x^{2}}{4}}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle x\;{}_{1}F_{2}\left({\frac {1}{2}};{\frac {3}{2}},{\frac {3}{2}};-{\frac {x^{2}}{4}}\right)}</annotation>

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</math><img src="./8027d2d9905d46716a9747e38f8176c4d4c27938.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.512ex; height:6.343ex;" alt="{\displaystyle x\;{}_{1}F_{2}\left({\frac {1}{2}};{\frac {3}{2}},{\frac {3}{2}};-{\frac {x^{2}}{4}}\right)}" loading="lazy"> is the antiderivative of the <a href="Cardinal_sine" class="mw-redirect" title="Cardinal sine">cardinal sine</a>. With modified values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}">

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

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</math><img src="./bbf42ecda092975c9c69dae84e16182ba5fe2e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1}}">

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

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</math><img src="./9af2720c91be489f57ecde4bb651b95e113d0144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{1}}" loading="lazy">, one obtains the antiderivative of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x^{\beta })/x^{\alpha }}">

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<annotation encoding="application/x-tex">{\displaystyle \sin(x^{\beta })/x^{\alpha }}</annotation>

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</math><img src="./5812f47700dfe69d02180102803a73e29d3e967e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.945ex; height:3.176ex;" alt="{\displaystyle \sin(x^{\beta })/x^{\alpha }}" loading="lazy">.<a href="#cite_note-10">[10]</a>

The <a href="Lommel_function" title="Lommel function">Lommel function</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{\mu ,\nu }(z)={\frac {z^{\mu +1}}{(\mu -\nu +1)(\mu +\nu +1)}}{}_{1}F_{2}(1;{\frac {\mu }{2}}-{\frac {\nu }{2}}+{\frac {3}{2}},{\frac {\mu }{2}}+{\frac {\nu }{2}}+{\frac {3}{2}};-{\frac {z^{2}}{4}})}">

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<annotation encoding="application/x-tex">{\displaystyle s_{\mu ,\nu }(z)={\frac {z^{\mu +1}}{(\mu -\nu +1)(\mu +\nu +1)}}{}_{1}F_{2}(1;{\frac {\mu }{2}}-{\frac {\nu }{2}}+{\frac {3}{2}},{\frac {\mu }{2}}+{\frac {\nu }{2}}+{\frac {3}{2}};-{\frac {z^{2}}{4}})}</annotation>

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</math><img src="./add58067a54503a96076323014a79d152e6f0fc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:71.484ex; height:6.509ex;" alt="{\displaystyle s_{\mu ,\nu }(z)={\frac {z^{\mu +1}}{(\mu -\nu +1)(\mu +\nu +1)}}{}_{1}F_{2}(1;{\frac {\mu }{2}}-{\frac {\nu }{2}}+{\frac {3}{2}},{\frac {\mu }{2}}+{\frac {\nu }{2}}+{\frac {3}{2}};-{\frac {z^{2}}{4}})}" loading="lazy">.<a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading3"><h3 id="The_series_2F0">The series 2F0</h3></div>

The confluent hypergeometric function of the second kind can be written as:<a href="#cite_note-12">[12]</a>

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<annotation encoding="application/x-tex">{\displaystyle U(a,b,z)=z^{-a}\;{}_{2}F_{0}\left(a,a-b+1;;-{\frac {1}{z}}\right).}</annotation>

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</math><img src="./6e57e2e8979339a251d7be5d4217ee22721c903f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.844ex; height:6.176ex;" alt="{\displaystyle U(a,b,z)=z^{-a}\;{}_{2}F_{0}\left(a,a-b+1;;-{\frac {1}{z}}\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="The_series_2F1">The series 2F1</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Hypergeometric_function" title="Hypergeometric function">Hypergeometric function</a></div>

Historically, the most important are the functions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{2}F_{1}(a,b;c;z)}">

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<annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}(a,b;c;z)}</annotation>

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</math><img src="./408c256b938cb39368545cdc782f73c0b5f791ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.837ex; height:2.843ex;" alt="{\displaystyle {}_{2}F_{1}(a,b;c;z)}" loading="lazy">. These are sometimes called Gauss's hypergeometric functions, classical standard hypergeometric or often simply hypergeometric functions. The term Generalized hypergeometric function is used for the functions pFq if there is risk of confusion. This function was first studied in detail by <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>, who explored the conditions for its convergence.

The differential equation for this function is

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<annotation encoding="application/x-tex">{\displaystyle \left(z{\frac {d}{dz}}+a\right)\left(z{\frac {d}{dz}}+b\right)w=\left(z{\frac {d}{dz}}+c\right){\frac {dw}{dz}}}</annotation>

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</math><img src="./1147c0f83e360d9aef240bd865211f0633dd081a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.344ex; height:6.176ex;" alt="{\displaystyle \left(z{\frac {d}{dz}}+a\right)\left(z{\frac {d}{dz}}+b\right)w=\left(z{\frac {d}{dz}}+c\right){\frac {dw}{dz}}}" loading="lazy"></dd></dl>

or

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<annotation encoding="application/x-tex">{\displaystyle z(1-z){\frac {d^{2}w}{dz^{2}}}+\left[c-(a+b+1)z\right]{\frac {dw}{dz}}-ab\,w=0.}</annotation>

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</math><img src="./08e8e3756bb7c64f4d8d5bd0b3aea7b3890ec783.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:48.84ex; height:6.009ex;" alt="{\displaystyle z(1-z){\frac {d^{2}w}{dz^{2}}}+\left[c-(a+b+1)z\right]{\frac {dw}{dz}}-ab\,w=0.}" loading="lazy"></dd></dl>

It is known as the <a href="Hypergeometric_differential_equation" class="mw-redirect" title="Hypergeometric differential equation">hypergeometric differential equation</a>. When c is not a positive integer, the substitution

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<annotation encoding="application/x-tex">{\displaystyle w=z^{1-c}u}</annotation>

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</math><img src="./c36754c3a025f632af4249c2a5338a02531c5d83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.227ex; height:2.676ex;" alt="{\displaystyle w=z^{1-c}u}" loading="lazy"></dd></dl>

gives a linearly independent solution

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<annotation encoding="application/x-tex">{\displaystyle z^{1-c}\;{}_{2}F_{1}(1+a-c,1+b-c;2-c;z),}</annotation>

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</math><img src="./474270d5fa11132810c70afd7b4597223b3ea4aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.967ex; height:3.176ex;" alt="{\displaystyle z^{1-c}\;{}_{2}F_{1}(1+a-c,1+b-c;2-c;z),}" loading="lazy"></dd></dl>

so the general solution for |z| < 1 is

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<annotation encoding="application/x-tex">{\displaystyle k\;{}_{2}F_{1}(a,b;c;z)+lz^{1-c}\;{}_{2}F_{1}(1+a-c,1+b-c;2-c;z)}</annotation>

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</math><img src="./fbc3f4b5dfce015e6d9cfd24748277c3fb7bbf76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.546ex; height:3.176ex;" alt="{\displaystyle k\;{}_{2}F_{1}(a,b;c;z)+lz^{1-c}\;{}_{2}F_{1}(1+a-c,1+b-c;2-c;z)}" loading="lazy"></dd></dl>

where k, l are constants. Different solutions can be derived for other values of z. In fact there are 24 solutions, known as the <a href="Ernst_Kummer" title="Ernst Kummer">Kummer</a> solutions, derivable using various identities, valid in different regions of the complex plane.

When a is a non-positive integer, −n,

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<annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}(-n,b;c;z)}</annotation>

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</math><img src="./4002912f1dcabf90b513dbe1af83167001837953.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.81ex; height:2.843ex;" alt="{\displaystyle {}_{2}F_{1}(-n,b;c;z)}" loading="lazy"></dd></dl>

is a polynomial. Up to constant factors and scaling, these are the <a href="Jacobi_polynomials" title="Jacobi polynomials">Jacobi polynomials</a>. Several other classes of orthogonal polynomials, up to constant factors, are special cases of Jacobi polynomials, so these can be expressed using 2F1 as well. This includes <a href="Legendre_polynomial" class="mw-redirect" title="Legendre polynomial">Legendre polynomials</a> and <a href="Chebyshev_polynomial" class="mw-redirect" title="Chebyshev polynomial">Chebyshev polynomials</a>.

A wide range of integrals of elementary functions can be expressed using the hypergeometric function, e.g.:

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<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{x}{\sqrt {1+y^{\alpha }}}\,\mathrm {d} y={\frac {x}{2+\alpha }}\left\{\alpha \;{}_{2}F_{1}\left({\tfrac {1}{\alpha }},{\tfrac {1}{2}};1+{\tfrac {1}{\alpha }};-x^{\alpha }\right)+2{\sqrt {x^{\alpha }+1}}\right\},\qquad \alpha \neq 0.}</annotation>

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</math><img src="./07d26b7c0d8807dd7094c0c6355e51c43fa7966a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:79.161ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{x}{\sqrt {1+y^{\alpha }}}\,\mathrm {d} y={\frac {x}{2+\alpha }}\left\{\alpha \;{}_{2}F_{1}\left({\tfrac {1}{\alpha }},{\tfrac {1}{2}};1+{\tfrac {1}{\alpha }};-x^{\alpha }\right)+2{\sqrt {x^{\alpha }+1}}\right\},\qquad \alpha \neq 0.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="The_series_2F2">The series 2F2</h3></div>

The hypergeometric series <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}_{2}F_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{2}}</annotation>

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</math><img src="./e841dd1322872ac6e4f8e4d0fdd6da13397d81e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.603ex; height:2.509ex;" alt="{\displaystyle {}_{2}F_{2}}" loading="lazy"> is generally associated with integrals of products of power functions and the exponential function. As such, the <a href="Exponential_integral" title="Exponential integral">exponential integral</a> can be written as:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ei} (x)=x{}_{2}F_{2}(1,1;2,2;x)+\ln x+\gamma .}</annotation>

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</math><img src="./998fe446c74124a3efb8a8b3be575646313407e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.571ex; height:2.843ex;" alt="{\displaystyle \operatorname {Ei} (x)=x{}_{2}F_{2}(1,1;2,2;x)+\ln x+\gamma .}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="The_series_3F0">The series 3F0</h3></div>

The <a href="Mott_polynomials" title="Mott polynomials">Mott polynomials</a> can be written as:<a href="#cite_note-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle s_{n}(x)=(-x/2)^{n}{}_{3}F_{0}(-n,{\frac {1-n}{2}},1-{\frac {n}{2}};;-{\frac {4}{x^{2}}}).}</annotation>

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</math><img src="./f2da86e5018c0598efc1d8d620f43e6af903c791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:47.93ex; height:5.509ex;" alt="{\displaystyle s_{n}(x)=(-x/2)^{n}{}_{3}F_{0}(-n,{\frac {1-n}{2}},1-{\frac {n}{2}};;-{\frac {4}{x^{2}}}).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="The_series_3F2">The series 3F2</h3></div>

The function

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Li} _{2}(x)=\sum _{n>0}\,{x^{n}}{n^{-2}}=x\;{}_{3}F_{2}(1,1,1;2,2;x)}</annotation>

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</math><img src="./de590c74e8c3f0af7323b5b52502b58b81347bf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.593ex; height:5.509ex;" alt="{\displaystyle \operatorname {Li} _{2}(x)=\sum _{n>0}\,{x^{n}}{n^{-2}}=x\;{}_{3}F_{2}(1,1,1;2,2;x)}" loading="lazy"></dd></dl></dd></dl>

is the <a href="Dilogarithm" title="Dilogarithm">dilogarithm</a><a href="#cite_note-14">[14]</a>

The function

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<annotation encoding="application/x-tex">{\displaystyle Q_{n}(x;a,b,N)={}_{3}F_{2}(-n,-x,n+a+b+1;a+1,-N+1;1)}</annotation>

</semantics>

</math><img src="./13b610210951093197def578221d4b2c98854104.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.185ex; height:2.843ex;" alt="{\displaystyle Q_{n}(x;a,b,N)={}_{3}F_{2}(-n,-x,n+a+b+1;a+1,-N+1;1)}" loading="lazy"></dd></dl></dd></dl>

is a <a href="Hahn_polynomial" class="mw-redirect" title="Hahn polynomial">Hahn polynomial</a>.

<div class="mw-heading mw-heading3"><h3 id="The_series_4F3">The series 4F3</h3></div>

The function

<msub>

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<annotation encoding="application/x-tex">{\displaystyle p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}\;{}_{4}F_{3}\left(-n,a+b+c+d+n-1,a-t,a+t;a+b,a+c,a+d;1\right)}</annotation>

</semantics>

</math><img src="./85233f657b0d6cd99b47990fae936e85ded2283e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:98.3ex; height:3.176ex;" alt="{\displaystyle p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}\;{}_{4}F_{3}\left(-n,a+b+c+d+n-1,a-t,a+t;a+b,a+c,a+d;1\right)}" loading="lazy"></dd></dl></dd></dl>

is a <a href="Wilson_polynomial" class="mw-redirect" title="Wilson polynomial">Wilson polynomial</a>.

All roots of a <a href="Quintic_equation" class="mw-redirect" title="Quintic equation">quintic equation</a> can be expressed in terms of radicals and the <a href="Bring_radical" title="Bring radical">Bring radical</a>, which is the real solution to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{5}+x+a=0}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle x^{5}+x+a=0}</annotation>

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</math><img src="./24f55168924a2a25fd0a2e35e2f08e03c4399a1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.885ex; height:2.843ex;" alt="{\displaystyle x^{5}+x+a=0}" loading="lazy">. The Bring radical can be written as:<a href="#cite_note-BR-15">[15]</a>

<msub>

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<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {BR} (a)=-a\;{}_{4}F_{3}\left({\frac {1}{5}},{\frac {2}{5}},{\frac {3}{5}},{\frac {4}{5}};{\frac {1}{2}},{\frac {3}{4}},{\frac {5}{4}};{\frac {3125a^{4}}{256}}\right).}</annotation>

</semantics>

</math><img src="./eb596a6e7798eec1d17a518a785c95beb54a5c95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.62ex; height:6.343ex;" alt="{\displaystyle \operatorname {BR} (a)=-a\;{}_{4}F_{3}\left({\frac {1}{5}},{\frac {2}{5}},{\frac {3}{5}},{\frac {4}{5}};{\frac {1}{2}},{\frac {3}{4}},{\frac {5}{4}};{\frac {3125a^{4}}{256}}\right).}" loading="lazy"></dd></dl></dd></dl>

The partition function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(K)}">

</mstyle>

</mrow>

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

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</math><img src="./4893515dcb866a2b38862c8267a4a1741db3ba4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.556ex; height:2.843ex;" alt="{\displaystyle Z(K)}" loading="lazy"> of the 2D isotropic <a href="Square_lattice_Ising_model" title="Square lattice Ising model">Ising model</a> with no external magnetic field was found by <a href="Lars_Onsager" title="Lars Onsager">Onsager</a> in the 1940s and can be expressed as<a href="#cite_note-Ising-16">[16]</a>

<msup>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \ln Z(K)=\ln(2\cosh 2K)-k^{2}{}_{4}F_{3}\left(1,1,{\frac {3}{2}},{\frac {3}{2}};2,2,2;16k^{2}\right),}</annotation>

</semantics>

</math><img src="./f1603bcfb8cdf5bde9d2dd0f04140d049aef41d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.486ex; height:6.176ex;" alt="{\displaystyle \ln Z(K)=\ln(2\cosh 2K)-k^{2}{}_{4}F_{3}\left(1,1,{\frac {3}{2}},{\frac {3}{2}};2,2,2;16k^{2}\right),}" loading="lazy"></dd></dl></dd></dl>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {J}{k_{\mathrm {B} }T}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle K={\frac {J}{k_{\mathrm {B} }T}}}</annotation>

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</math><img src="./f652146d599183b56925d2b7425880aff2839dc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.244ex; height:5.676ex;" alt="{\displaystyle K={\frac {J}{k_{\mathrm {B} }T}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\frac {1}{2}}\tanh 2K\,\operatorname {sech} 2K}">

<mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k={\frac {1}{2}}\tanh 2K\,\operatorname {sech} 2K}</annotation>

</semantics>

</math><img src="./4fc4a657993c028de2e7a750b11b5051b43e6278.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.626ex; height:5.176ex;" alt="{\displaystyle k={\frac {1}{2}}\tanh 2K\,\operatorname {sech} 2K}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="The_series_q+1Fq">The series q+1Fq</h3></div>

The functions

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Li} _{q}(z)=z\;{}_{q+1}F_{q}\left(1,1,\ldots ,1;2,2,\ldots ,2;z\right)}</annotation>

</semantics>

</math><img src="./99f63fcc38872f2f310a69af820c7cd016b0a0ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:41.141ex; height:3.009ex;" alt="{\displaystyle \operatorname {Li} _{q}(z)=z\;{}_{q+1}F_{q}\left(1,1,\ldots ,1;2,2,\ldots ,2;z\right)}" loading="lazy"></dd>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Li} _{-p}(z)=z\;{}_{p}F_{p-1}\left(2,2,\ldots ,2;1,1,\ldots ,1;z\right)}</annotation>

</semantics>

</math><img src="./53f264a5d99357cdc4008db05df849ef2a13da27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.631ex; height:3.009ex;" alt="{\displaystyle \operatorname {Li} _{-p}(z)=z\;{}_{p}F_{p-1}\left(2,2,\ldots ,2;1,1,\ldots ,1;z\right)}" loading="lazy"></dd></dl></dd></dl>

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\in \mathbb {N} _{0}}">

<msub>

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</mrow>

</msub>

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<annotation encoding="application/x-tex">{\displaystyle q\in \mathbb {N} _{0}}</annotation>

</semantics>

</math><img src="./bd60423a116999d57897b700f035cef944e2e579.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.643ex; height:2.509ex;" alt="{\displaystyle q\in \mathbb {N} _{0}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./7d4dd27ec9da79b60215c701fa49fb4c5af302b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.778ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {N} }" loading="lazy"> are the <a href="Polylogarithm" title="Polylogarithm">Polylogarithm</a>.

For each integer n≥2, the roots of the polynomial xn−x+t can be expressed as a sum of at most N−1 hypergeometric functions of type n+1Fn, which can always be reduced by eliminating at least one pair of a and b parameters.<a href="#cite_note-BR-15">[15]</a>

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

The generalized hypergeometric function is linked to the <a href="Meijer_G-function" title="Meijer G-function">Meijer G-function</a> and the <a href="MacRobert_E-function" class="mw-redirect" title="MacRobert E-function">MacRobert E-function</a>. Hypergeometric series were generalised to several variables, for example by <a href="Paul_Emile_Appell" class="mw-redirect" title="Paul Emile Appell">Paul Emile Appell</a> and <a href="Joseph_Kamp%C3%A9_de_F%C3%A9riet" title="Joseph Kampé de Fériet">Joseph Kampé de Fériet</a>; but a comparable general theory took long to emerge. Many identities were found, some quite remarkable. A generalization, the <a href="Q-series" class="mw-redirect" title="Q-series">q-series</a> analogues, called the <a href="Basic_hypergeometric_series" title="Basic hypergeometric series">basic hypergeometric series</a>, were given by <a href="Eduard_Heine" title="Eduard Heine">Eduard Heine</a> in the late nineteenth century. Here, the ratios considered of successive terms, instead of a rational function of n, are a rational function of qn. Another generalization, the <a href="Elliptic_hypergeometric_series" title="Elliptic hypergeometric series">elliptic hypergeometric series</a>, are those series where the ratio of terms is an <a href="Elliptic_function" title="Elliptic function">elliptic function</a> (a doubly periodic <a href="Meromorphic_function" title="Meromorphic function">meromorphic function</a>) of n.

During the twentieth century this was a fruitful area of combinatorial mathematics, with numerous connections to other fields. There are a number of new definitions of <a href="General_hypergeometric_function" title="General hypergeometric function">general hypergeometric functions</a>, by Aomoto, <a href="Israel_Gelfand" title="Israel Gelfand">Israel Gelfand</a> and others; and applications for example to the combinatorics of arranging a number of <a href="Hyperplane" title="Hyperplane">hyperplanes</a> in complex N-space (see <a href="Arrangement_of_hyperplanes" title="Arrangement of hyperplanes">arrangement of hyperplanes</a>).

Special hypergeometric functions occur as <a href="Zonal_spherical_function" title="Zonal spherical function">zonal spherical functions</a> on <a href="Symmetric_space" title="Symmetric space">Riemannian symmetric spaces</a> and semi-simple <a href="Lie_group" title="Lie group">Lie groups</a>. Their importance and role can be understood through the following example: the hypergeometric series 2F1 has the <a href="Legendre_polynomials" title="Legendre polynomials">Legendre polynomials</a> as a special case, and when considered in the form of <a href="Spherical_harmonics" title="Spherical harmonics">spherical harmonics</a>, these polynomials reflect, in a certain sense, the symmetry properties of the two-sphere or, equivalently, the rotations given by the Lie group <a href="SO(3)" class="mw-redirect" title="SO(3)">SO(3)</a>. In tensor product decompositions of concrete representations of this group <a href="Clebsch%E2%80%93Gordan_coefficients" title="Clebsch–Gordan coefficients">Clebsch–Gordan coefficients</a> are met, which can be written as 3F2 hypergeometric series.

<a href="Bilateral_hypergeometric_series" title="Bilateral hypergeometric series">Bilateral hypergeometric series</a> are a generalization of hypergeometric functions where one sums over all integers, not just the positive ones.

<a href="Fox%E2%80%93Wright_function" title="Fox–Wright function">Fox–Wright functions</a> are a generalization of generalized hypergeometric functions where the Pochhammer symbols in the series expression are generalised to gamma functions of linear expressions in the index n.

<ul><li><a href="Appell_series" title="Appell series">Appell series</a></li>

<li><a href="Humbert_series" title="Humbert series">Humbert series</a></li>

<li><a href="Kamp%C3%A9_de_F%C3%A9riet_function" title="Kampé de Fériet function">Kampé de Fériet function</a></li>

<li><a href="Lauricella_hypergeometric_series" title="Lauricella hypergeometric series">Lauricella hypergeometric series</a></li></ul>

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

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</style><cite id="CITEREFPrudnikovBrychkovMarichev1990" class="citation book cs1">Prudnikov, A. P.; Brychkov, Yu. A.; Marichev, O. I. (1990). Integrals & Series Volume 3: More Special Functions. Gordon and Breach. p. 439.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFKarlsson1970" class="citation journal cs1">Karlsson, Per W. (1970). <a rel="nofollow" class="external text" href="https://backend.orbit.dtu.dk/ws/files/3645014/Per.pdf">"Hypergeometric functions with integral parameter differences"</a> (PDF). J. Math. Phys. 12 (2): 270–271. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1665587">10.1063/1.1665587</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> (<a href="#CITEREFSlater1966">Slater 1966</a>, Equation (4.1.2))

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFGottschalkMaslen1988" class="citation journal cs1">Gottschalk, J. E.; Maslen, E. N. (1988). "Reduction formulae for generalised hypergeometric functions of one variable". J. Phys. A: Math. Gen. 21 (9): 1983–1998. <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/1988JPhA...21.1983G">1988JPhA...21.1983G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0305-4470%2F21%2F9%2F015">10.1088/0305-4470/21/9/015</a>.</cite>

</li>

<cite id="CITEREFRainville1945" class="citation journal cs1">Rainville, D. (1945). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1945-08425-0">"The contiguous function relations for pFq with application to Bateman's J and Rice's H"</a>. Bull. Amer. Math. Soc. 51 (10): 714–723. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1945-08425-0">10.1090/S0002-9904-1945-08425-0</a>.</cite>

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> See (<a href="#CITEREFSlater1966">Slater 1966</a>, Section 2.3.1) or (<a href="#CITEREFBailey1935">Bailey 1935</a>, Section 2.2) for a proof, or the <a href="https://proofwiki.org/wiki/Pfaff-Saalsch%C3%BCtz_Theorem" class="extiw external" title="proofwiki:Pfaff-Saalschütz Theorem">ProofWiki</a>.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Pfaff, J. F. [1797]. Observations analyticae ad L. Euleri Institutiones Calculi Integralis.  Vol. IV, Supplem. II et IV, Historie de 1793, Nova Acata Acad. Scie. Petropolitanae.  XI, 38-57. (Note: The history section is paged separately from the scientific section  of this journal.)

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> See (<a href="#CITEREFBailey1935">Bailey 1935</a>, Section 3.1) for a detailed proof. An alternative proof is in (<a href="#CITEREFSlater1966">Slater 1966</a>, Section 2.3.3)

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Dougall-RamanujanIdentity.html">"Dougall-Ramanujan Identity"</a>. mathworld.wolfram.com. Retrieved 2025-03-13.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> Victor Nijimbere, Ural Math J vol 3(1) and <a rel="nofollow" class="external free" href="https://arxiv.org/abs/1703.01907">https://arxiv.org/abs/1703.01907</a> (2017)

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dlmf.nist.gov/13.6">"DLMF: §13.6 Relations to Other Functions ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions"</a>. dlmf.nist.gov.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> See Erdélyi et al. 1955.

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFCandan" class="citation web cs1">Candan, Cagatay. <a rel="nofollow" class="external text" href="http://www.eee.metu.edu.tr/~ccandan/pub_dir/hyper_rel.pdf">"A Simple Proof of F(1,1,1;2,2;x)=dilog(1-x)/x"</a> (PDF).</cite>

</li>

<cite id="CITEREFGlasser1994" class="citation arxiv cs1">Glasser, M. Lawrence (1994). "The quadratic formula made hard: A less radical approach to solving equations". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.CA/9411224">math.CA/9411224</a>.</cite>

</li>

<cite id="CITEREFViswanathan2014" class="citation journal cs1">Viswanathan, G. M. (2014). "The hypergeometric series for the partition function of the 2-D Ising model". Journal of Statistical Mechanics: Theory and Experiment. 2015 (7): 07004. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1411.2495">1411.2495</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/2015JSMTE..07..004V">2015JSMTE..07..004V</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1742-5468%2F2015%2F07%2FP07004">10.1088/1742-5468/2015/07/P07004</a>.</cite>

</li>

</ol></div>

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

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<li><cite id="CITEREFAndrewsAskeyRoy1999" class="citation book cs1">Andrews, George E.; Askey, Richard & Roy, Ranjan (1999). Special functions. Encyclopedia of Mathematics and its Applications. Vol. 71. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-78988-2</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=1688958">1688958</a>.</cite></li>

<li><cite id="CITEREFBailey1935" class="citation book cs1">Bailey, W.N. (1935). Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 32. London: Cambridge University Press. <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:0011.02303">0011.02303</a>.</cite></li>

<li><cite id="CITEREFDixon1902" class="citation journal cs1">Dixon, A.C. (1902). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1433433">"Summation of a certain series"</a>. Proc. London Math. Soc. 35 (1): 284–291. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs1-35.1.284">10.1112/plms/s1-35.1.284</a>. <a href="JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:34.0490.02">34.0490.02</a>.</cite></li>

<li><cite id="CITEREFDougall1907" class="citation journal cs1">Dougall, J. (1907). <a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0013091500033642">"On Vandermonde's theorem and some more general expansions"</a>. Proc. Edinburgh Math. Soc. 25: 114–132. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0013091500033642">10.1017/S0013091500033642</a>.</cite></li>

<li><cite id="CITEREFErdélyiMagnusOberhettingerTricomi1955" class="citation book cs1">Erdélyi, Arthur; <a href="Wilhelm_Magnus" title="Wilhelm Magnus">Magnus, Wilhelm</a>; Oberhettinger, Fritz; Tricomi, Francesco G. (1955). Higher transcendental functions. Vol. III. McGraw-Hill Book Company, Inc., New York-Toronto-London. <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=0066496">0066496</a>.</cite></li>

<li><cite id="CITEREFGasperRahman2004" class="citation book cs1">Gasper, George; <a href="Mizan_Rahman" title="Mizan Rahman">Rahman, Mizan</a> (2004). Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Applications. Vol. 96 (2nd ed.). Cambridge, UK: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-83357-8</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=2128719">2128719</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:1129.33005">1129.33005</a>.</cite> (the first edition has <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-35049-2</bdi>)</li>

<li><cite id="CITEREFGauss1813" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss, Carl Friedrich</a> (1813). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=uDMAAAAAQAAJ">"Disquisitiones generales circa seriam infinitam   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\tfrac {\alpha \beta }{1\cdot \gamma }}~x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}~x~x+{\mbox{etc.}}}">

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<annotation encoding="application/x-tex">{\displaystyle 1+{\tfrac {\alpha \beta }{1\cdot \gamma }}~x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}~x~x+{\mbox{etc.}}}</annotation>

</semantics>

</math><img src="./5351a876f44e876445e2481dcdbea2ecd91fec97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.622ex; height:4.843ex;" alt="{\displaystyle 1+{\tfrac {\alpha \beta }{1\cdot \gamma }}~x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}~x~x+{\mbox{etc.}}}" loading="lazy">"</a>. Commentationes Societatis Regiae Scientarum Gottingensis Recentiores (in Latin). 2. Göttingen.</cite> (a reprint of this paper can be found in <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_uDMAAAAAQAAJ">Carl Friedrich Gauss, Werke</a>, p. 125) (a translation is available <a href="https://en.wikisource.org/wiki/Translation:Disquisitiones_generales_circa_seriem_infinitam_..." class="extiw external" title="wikisource:Translation:Disquisitiones generales circa seriem infinitam ...">on Wikisource</a>)</li>

<li><cite id="CITEREFGrinshpan2013" class="citation cs2">Grinshpan, A. Z. (2013), "Generalized hypergeometric functions: product identities and weighted norm inequalities", The Ramanujan Journal, 31 (1–2): 53–66, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs11139-013-9487-x">10.1007/s11139-013-9487-x</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121054930">121054930</a></cite></li></ul>

<ul><li><cite id="CITEREFHeckmanSchlichtkrull1994" class="citation book cs1">Heckman, Gerrit & Schlichtkrull, Henrik (1994). Harmonic Analysis and Special Functions on Symmetric Spaces. San Diego: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-336170-7</bdi>.</cite> (part 1 treats hypergeometric functions on Lie groups)</li>

<li><cite id="CITEREFLavoieGrondinRathieArora1994" class="citation journal cs1">Lavoie, J.L.; Grondin, F.; Rathie, A.K.; Arora, K. (1994). "Generalizations of Dixon's theorem on the sum of a 3F2". Math. Comp. 62 (205): 267–276. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2153407">10.2307/2153407</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2153407">2153407</a>.</cite></li>

<li><cite id="CITEREFMillerParis2011" class="citation journal cs1">Miller, A. R.; Paris, R. B. (2011). <a rel="nofollow" class="external text" href="https://rke.abertay.ac.uk/en/publications/30e4ad50-271e-40a7-bfb3-dc6515871b50">"Euler-type transformations for the generalized hypergeometric function r+2Fr+1"</a>. Z. Angew. Math. Phys. 62 (1): 31–45. <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/2011ZaMP...62...31M">2011ZaMP...62...31M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs00033-010-0085-0">10.1007/s00033-010-0085-0</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:30484300">30484300</a>.</cite></li>

<li><cite id="CITEREFQuigleyWilsonWallsBedford2013" class="citation journal cs1">Quigley, J.; Wilson, K.J.; Walls, L.; Bedford, T. (2013). <a rel="nofollow" class="external text" href="https://strathprints.strath.ac.uk/43403/1/EventRatesRA.pdf">"A Bayes linear Bayes Method for Estimation of Correlated Event Rates"</a> (PDF). Risk Analysis. 33 (12): 2209–2224. <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/2013RiskA..33.2209Q">2013RiskA..33.2209Q</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Frisa.12035">10.1111/risa.12035</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/23551053">23551053</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:24476762">24476762</a>.</cite></li>

<li><cite id="CITEREFRathiePogány2008" class="citation journal cs1">Rathie, Arjun K.; Pogány, Tibor K. (2008). <a rel="nofollow" class="external text" href="http://hrcak.srce.hr/file/37118">"New summation formula for 3F2(1/2) and a Kummer-type II transformation of 2F2(x)"</a>. Mathematical Communications. 13: 63–66. <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=2422088">2422088</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:1146.33002">1146.33002</a>.</cite></li>

<li><cite id="CITEREFRakhaRathie2011" class="citation journal cs1">Rakha, M.A.; Rathie, Arjun K. (2011). <a rel="nofollow" class="external text" href="https://doi.org/10.4134%2Fbkms.2011.48.1.151">"Extensions of Euler's type- II transformation and Saalschutz's theorem"</a>. Bull. Korean Math. Soc. 48 (1): 151–156. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4134%2Fbkms.2011.48.1.151">10.4134/bkms.2011.48.1.151</a>.</cite></li>

<li><cite id="CITEREFSaalschütz1890" class="citation journal cs1 cs1-prop-foreign-lang-source">Saalschütz, L. (1890). "Eine Summationsformel". Zeitschrift für Mathematik und Physik (in German). 35: 186–188. <a href="JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:22.0262.03">22.0262.03</a>.</cite></li>

<li><cite id="CITEREFSlater1966" class="citation book cs1"><a href="Lucy_Joan_Slater" title="Lucy Joan Slater">Slater, Lucy Joan</a> (1966). Generalized Hypergeometric Functions. Cambridge, UK: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-06483-5</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=0201688">0201688</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:0135.28101">0135.28101</a>.</cite> (there is a 2008 paperback with <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-09061-2</bdi>)</li>

<li><cite id="CITEREFYoshida1997" class="citation book cs1">Yoshida, Masaaki (1997). Hypergeometric Functions, My Love: Modular Interpretations of Configuration Spaces. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-528-06925-4</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=1453580">1453580</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="https://web.archive.org/web/20060129095451/http://www.cis.upenn.edu/~wilf/AeqB.html">The book "A = B"</a>, this book is freely downloadable from the internet.</li>

<li><a href="MathWorld" title="MathWorld">MathWorld</a>

<ul><li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/GeneralizedHypergeometricFunction.html">"Generalized Hypergeometric Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/HypergeometricFunction.html">"Hypergeometric Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ConfluentHypergeometricFunctionoftheFirstKind.html">"Confluent Hypergeometric Function of the First Kind"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ConfluentHypergeometricLimitFunction.html">"Confluent Hypergeometric Limit Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul></li></ul>

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</style><div id="Sequences_and_series332" style="font-size:114%;margin:0 4em"><a href="Sequence" title="Sequence">Sequences</a> and <a href="Series_(mathematics)" title="Series (mathematics)">series</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Integer_sequence" title="Integer sequence">Integer sequences</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Basic</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetic_progression" title="Arithmetic progression">Arithmetic progression</a></li>

<li><a href="Geometric_progression" title="Geometric progression">Geometric progression</a></li>

<li><a href="Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">Harmonic progression</a></li>

<li><a href="Square_number" title="Square number">Square number</a></li>

<li><a href="Cube_(algebra)" title="Cube (algebra)">Cubic number</a></li>

<li><a href="Factorial" title="Factorial">Factorial</a></li>

<li><a href="Power_of_two" title="Power of two">Powers of two</a></li>

<li><a href="Power_of_three" title="Power of three">Powers of three</a></li>

<li><a href="Power_of_10" title="Power of 10">Powers of 10</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Advanced (<a href="List_of_OEIS_sequences" class="mw-redirect" title="List of OEIS sequences">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Complete_sequence" title="Complete sequence">Complete sequence</a></li>

<li><a href="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a></li>

<li><a href="Figurate_number" title="Figurate number">Figurate number</a></li>

<li><a href="Heptagonal_number" title="Heptagonal number">Heptagonal number</a></li>

<li><a href="Hexagonal_number" title="Hexagonal number">Hexagonal number</a></li>

<li><a href="Lucas_number" title="Lucas number">Lucas number</a></li>

<li><a href="Pell_number" title="Pell number">Pell number</a></li>

<li><a href="Pentagonal_number" title="Pentagonal number">Pentagonal number</a></li>

<li><a href="Polygonal_number" title="Polygonal number">Polygonal number</a></li>

<li><a href="Triangular_number" title="Triangular number">Triangular number</a>

<ul><li><a href="Triangular_array" title="Triangular array">array</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><a href="Fibonacci_sequence" title="Fibonacci sequence"></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of sequences</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li>

<li><a href="Monotonic_function" title="Monotonic function">Monotonic function</a></li>

<li><a href="Periodic_sequence" title="Periodic sequence">Periodic sequence</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Series</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alternating_series" title="Alternating series">Alternating</a></li>

<li><a href="Convergent_series" title="Convergent series">Convergent</a></li>

<li><a href="Divergent_series" title="Divergent series">Divergent</a></li>

<li><a href="Telescoping_series" title="Telescoping series">Telescoping</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergence</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Absolute_convergence" title="Absolute convergence">Absolute</a></li>

<li><a href="Conditional_convergence" title="Conditional convergence">Conditional</a></li>

<li><a href="Uniform_convergence" title="Uniform convergence">Uniform</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicit series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergent</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Riemann_zeta_function" title="Riemann zeta function">1 + 1/2s + 1/3s + ... (Riemann zeta function)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Divergent</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Grandi's_series" title="Grandi's series">1 − 1 + 1 − 1 + ⋯ (Grandi's series)</a></li>

<li><a href="Infinite_arithmetic_series" class="mw-redirect" title="Infinite arithmetic series">Infinite arithmetic series</a></li>

<li><a href="1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_%E2%8B%AF" title="1 − 1 + 2 − 6 + 24 − 120 + ⋯">1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)</a></li>

<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)</a></li>

<li><a href="Divergence_of_the_sum_of_the_reciprocals_of_the_primes" title="Divergence of the sum of the reciprocals of the primes">1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Kinds of series</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Taylor_series" title="Taylor series">Taylor series</a></li>

<li><a href="Power_series" title="Power series">Power series</a></li>

<li><a href="Formal_power_series" title="Formal power series">Formal power series</a></li>

<li><a href="Laurent_series" title="Laurent series">Laurent series</a></li>

<li><a href="Puiseux_series" title="Puiseux series">Puiseux series</a></li>

<li><a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a></li>

<li><a href="Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li>

<li><a href="Fourier_series" title="Fourier series">Fourier series</a></li>

<li><a href="Generating_series" class="mw-redirect" title="Generating series">Generating series</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Hypergeometric_function" title="Hypergeometric function">Hypergeometric series</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">Generalized hypergeometric series</a></li>

<li><a href="Hypergeometric_function_of_a_matrix_argument" title="Hypergeometric function of a matrix argument">Hypergeometric function of a matrix argument</a></li>

<li><a href="Lauricella_hypergeometric_series" title="Lauricella hypergeometric series">Lauricella hypergeometric series</a></li>

<li><a href="Modular_hypergeometric_series" class="mw-redirect" title="Modular hypergeometric series">Modular hypergeometric series</a></li>

<li><a href="Riemann's_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li>

<li><a href="Theta_hypergeometric_series" class="mw-redirect" title="Theta hypergeometric series">Theta hypergeometric series</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div>

<ul><li> Category</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-31" href="https://en.wikipedia.org/wiki/?title=Generalized_hypergeometric_function&oldid=1303592966">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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