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Hypergeometric function

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Hypergeometric_distribution" title="Hypergeometric distribution">Hypergeometric distribution</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">The term "hypergeometric function" sometimes refers to the <a href="Generalized_hypergeometric_function" title="Generalized hypergeometric function">generalized hypergeometric function</a>. For other hypergeometric functions see <a href="#See_also">See also</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a <a href="Special_functions" title="Special functions">special function</a> represented by the hypergeometric series, that includes many other special functions as <a href="Special_case" title="Special case">specific</a> or <a href="Limiting_case_(mathematics)" title="Limiting case (mathematics)">limiting cases</a>. It is a solution of a second-order <a href="Linear_function" title="Linear function">linear</a> <a href="Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</a> (ODE). Every second-order linear ODE with three <a href="Regular_singular_point" title="Regular singular point">regular singular points</a> can be transformed into this equation.

For systematic lists of some of the many thousands of published <a href="Identity_(mathematics)" title="Identity (mathematics)">identities</a> involving the hypergeometric function, see the reference works by <a href="#CITEREFErdélyiMagnusOberhettingerTricomi1953">Erdélyi et al. (1953)</a> and <a href="#CITEREFOlde_Daalhuis2010">Olde Daalhuis (2010)</a>. There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic.

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

The term "hypergeometric series" was first used by <a href="John_Wallis" title="John Wallis">John Wallis</a> in his 1655 book Arithmetica Infinitorum.

Hypergeometric series were studied by <a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, but the first full systematic treatment was given by <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> (<a href="#CITEREFGauss1813">1813</a>).

Studies in the nineteenth century included those of <a href="Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a> (<a href="#CITEREFKummer1836">1836</a>), and the fundamental characterisation by <a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> (<a href="#CITEREFRiemann1857">1857</a>) of the hypergeometric function by means of the differential equation it satisfies.

Riemann showed that the second-order differential equation for 2F1(z), examined in the complex plane, could be characterised (on the <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a>) by its three <a href="Regular_singularity" class="mw-redirect" title="Regular singularity">regular singularities</a>.

The cases where the solutions are <a href="Algebraic_function" title="Algebraic function">algebraic functions</a> were found by <a href="Hermann_Schwarz" title="Hermann Schwarz">Hermann Schwarz</a> (<a href="Schwarz's_list" title="Schwarz's list">Schwarz's list</a>).

<div class="mw-heading mw-heading2"><h2 id="The_hypergeometric_series">The hypergeometric series</h2></div>

The hypergeometric function is defined for |z| < 1 by the <a href="Power_series" title="Power series">power series</a>

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<annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}(a,b;c;z)=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}}=1+{\frac {ab}{c}}{\frac {z}{1!}}+{\frac {a(a+1)b(b+1)}{c(c+1)}}{\frac {z^{2}}{2!}}+\cdots .}</annotation>

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It is undefined (or infinite) if c equals a non-positive <a href="Integer" title="Integer">integer</a>. Here (q)n is the (rising) <a href="Pochhammer_symbol" class="mw-redirect" title="Pochhammer symbol">Pochhammer symbol</a>,<a href="#cite_note-1">[note 1]</a> which is defined by:

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<annotation encoding="application/x-tex">{\displaystyle (q)_{n}={\begin{cases}1&n=0\\q(q+1)\cdots (q+n-1)&n>0\end{cases}}}</annotation>

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The series terminates if either a or b is a nonpositive integer, in which case the function reduces to a polynomial:

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<annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}(-m,b;c;z)=\sum _{n=0}^{m}(-1)^{n}{\binom {m}{n}}{\frac {(b)_{n}}{(c)_{n}}}z^{n}.}</annotation>

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For complex arguments z with |z| ≥ 1 it can be <a href="Analytic_continuation" title="Analytic continuation">analytically continued</a> along any path in the complex plane that avoids the branch points 1 and infinity. In practice, most computer implementations of the hypergeometric function adopt a branch cut along the line z ≥ 1.

As c → −m, where m is a non-negative integer, one has 2F1(z) → ∞. Dividing by the value Γ(c) of the <a href="Gamma_function" title="Gamma function">gamma function</a>, we have the limit:

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

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2F1(z) is the most common type of <a href="Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">generalized hypergeometric series</a> pFq, and is often designated simply F(z).

<div class="mw-heading mw-heading2"><h2 id="Differentiation_formulas">Differentiation formulas</h2></div>

Using the identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a)_{n+1}=a(a+1)_{n}}">

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

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</math><img src="./688b10013892901fee1768bf6b6a498f2d789c19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.947ex; height:2.843ex;" alt="{\displaystyle (a)_{n+1}=a(a+1)_{n}}" loading="lazy">, it is shown that

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

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and more generally,

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

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

Many of the common mathematical functions can be expressed in terms of the hypergeometric function, or as limiting cases of it. Some typical examples are

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<mtext> arbitrary</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}_{2}F_{1}\left(1,1;2;-z\right)&={\frac {\ln(1+z)}{z}}\\_{2}F_{1}(a,b;b;z)&=(1-z)^{-a}\quad (b{\text{ arbitrary}})\\_{2}F_{1}\left({\frac {1}{2}},{\frac {1}{2}};{\frac {3}{2}};z^{2}\right)&={\frac {\arcsin(z)}{z}}\\\,_{2}F_{1}\left({\frac {1}{3}},{\frac {2}{3}};{\frac {3}{2}};-{\frac {27x^{2}}{4}}\right)&={\frac {{\sqrt[{3}]{\frac {3x{\sqrt {3}}+{\sqrt {27x^{2}+4}}}{2}}}-{\sqrt[{3}]{\frac {2}{3x{\sqrt {3}}+{\sqrt {27x^{2}+4}}}}}}{x{\sqrt {3}}}}\\\end{aligned}}}</annotation>

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When a=1 and b=c, the series reduces into a plain <a href="Geometric_series" title="Geometric series">geometric series</a>, i.e.

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}_{2}F_{1}\left(1,b;b;z\right)&={}_{1}F_{0}\left(1;;z\right)=1+z+z^{2}+z^{3}+z^{4}+\cdots \end{aligned}}}</annotation>

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hence, the name hypergeometric. This function can be considered as a generalization of the <a href="Geometric_series" title="Geometric series">geometric series</a>.

The <a href="Confluent_hypergeometric_function" title="Confluent hypergeometric function">confluent hypergeometric function</a> (or Kummer's function) can be given as a limit of the hypergeometric function

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

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so all functions that are essentially special cases of it, such as <a href="Bessel_functions" class="mw-redirect" title="Bessel functions">Bessel functions</a>, can be expressed as limits of hypergeometric functions. These include most of the commonly used functions of mathematical physics.

<a href="Legendre_function" title="Legendre function">Legendre functions</a> are solutions of a second order differential equation with 3 regular singular points so can be expressed in terms of the hypergeometric function in many ways, for example

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

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Several orthogonal polynomials, including <a href="Jacobi_polynomials" title="Jacobi polynomials">Jacobi polynomials</a> P(α,β) n and their special cases <a href="Legendre_polynomials" title="Legendre polynomials">Legendre polynomials</a>, <a href="Chebyshev_polynomials" title="Chebyshev polynomials">Chebyshev polynomials</a>, <a href="Gegenbauer_polynomials" title="Gegenbauer polynomials">Gegenbauer polynomials</a>, <a href="Zernike_polynomials" title="Zernike polynomials">Zernike polynomials</a> can be written in terms of hypergeometric functions using

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<annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}(-n,\alpha +1+\beta +n;\alpha +1;x)={\frac {n!}{(\alpha +1)_{n}}}P_{n}^{(\alpha ,\beta )}(1-2x)}</annotation>

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Other polynomials that are special cases include <a href="Krawtchouk_polynomials" class="mw-redirect" title="Krawtchouk polynomials">Krawtchouk polynomials</a>, <a href="Meixner_polynomials" title="Meixner polynomials">Meixner polynomials</a>, <a href="Meixner%E2%80%93Pollaczek_polynomials" title="Meixner–Pollaczek polynomials">Meixner–Pollaczek polynomials</a>.

Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {C} \setminus \{0,1\}}">

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<annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {C} \setminus \{0,1\}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \tau ={\rm {i}}{\frac {{}_{2}F_{1}{\bigl (}{\frac {1}{2}},{\frac {1}{2}};1;1-z{\bigr )}}{{}_{2}F_{1}{\bigl (}{\frac {1}{2}},{\frac {1}{2}};1;z{\bigr )}}}.}</annotation>

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Then

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<annotation encoding="application/x-tex">{\displaystyle \lambda (\tau )={\frac {\theta _{2}(\tau )^{4}}{\theta _{3}(\tau )^{4}}}=z}</annotation>

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is the <a href="Modular_lambda_function" title="Modular lambda function">modular lambda function</a>, where

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<annotation encoding="application/x-tex">{\displaystyle \theta _{2}(\tau )=\sum _{n\in \mathbb {Z} }e^{\pi i\tau (n+1/2)^{2}},\quad \theta _{3}(\tau )=\sum _{n\in \mathbb {Z} }e^{\pi i\tau n^{2}}.}</annotation>

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The <a href="J-invariant" title="J-invariant">j-invariant</a>, a <a href="Modular_form#Modular_functions" title="Modular form">modular function</a>, is a rational function in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (\tau )}">

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

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<a href="Incomplete_beta_function" class="mw-redirect" title="Incomplete beta function">Incomplete beta functions</a> Bx(p,q) are related by

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<annotation encoding="application/x-tex">{\displaystyle B_{x}(p,q)={\tfrac {x^{p}}{p}}{}_{2}F_{1}(p,1-q;p+1;x).}</annotation>

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The <a href="Complete_elliptic_integral" class="mw-redirect" title="Complete elliptic integral">complete elliptic integrals</a> K and E are given by<a href="#cite_note-2">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K(k)&={\tfrac {\pi }{2}}\,_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1}{2}};1;k^{2}\right),\\E(k)&={\tfrac {\pi }{2}}\,_{2}F_{1}\left(-{\tfrac {1}{2}},{\tfrac {1}{2}};1;k^{2}\right).\end{aligned}}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="The_hypergeometric_differential_equation">The hypergeometric differential equation</h2></div>

The hypergeometric function is a solution of Euler's hypergeometric differential equation

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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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which has three <a href="Regular_singular_point" title="Regular singular point">regular singular points</a>: 0,1 and ∞. The generalization of this equation to three arbitrary regular singular points is given by <a href="Riemann's_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a>. Any second order linear differential equation with three regular singular points can be converted to the hypergeometric differential equation by a change of variables.

<div class="mw-heading mw-heading3"><h3 id="Solutions_at_the_singular_points">Solutions at the singular points</h3></div>

Solutions to the hypergeometric differential equation are built out of the hypergeometric series 2F1(a,b;c;z). The equation has two <a href="Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> solutions. At each of the three singular points 0, 1, ∞, there are usually two special solutions of the form xs times a holomorphic function of x, where s is one of the two roots of the indicial equation and x is a local variable vanishing at a regular singular point. This gives 3 × 2 = 6 special solutions, as follows.

Around the point z = 0, two independent solutions are, if c is not a non-positive integer,

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

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and, on condition that c is not an integer,

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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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If c is a non-positive integer 1−m, then the first of these solutions does not exist and must be replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{m}F(a+m,b+m;1+m;z).}">

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

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</math><img src="./9977d89e88e24439517dad53db7b143bde5cf7d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.184ex; height:2.843ex;" alt="{\displaystyle z^{m}F(a+m,b+m;1+m;z).}" loading="lazy"> The second solution does not exist when c is an integer greater than 1, and is equal to the first solution, or its replacement, when c is any other integer. So when c is an integer, a more complicated expression must be used for a second solution, equal to the first solution multiplied by ln(z), plus another series in powers of z, involving the <a href="Digamma_function" title="Digamma function">digamma function</a>. See <a href="#CITEREFOlde_Daalhuis2010">Olde Daalhuis (2010)</a> for details.

Around z = 1, if c − a − b is not an integer, one has two independent solutions

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

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and

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

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Around z = ∞, if a − b is not an integer, one has two independent solutions

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

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and

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

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Again, when the conditions of non-integrality are not met, there exist other solutions that are more complicated.

Any 3 of the above 6 solutions satisfy a linear relation as the space of solutions is 2-dimensional, giving (6 3) = 20 linear relations between them called connection formulas.

<div class="mw-heading mw-heading3"><h3 id="Kummer's_24_solutions">Kummer's 24 solutions</h3></div>

A second order <a href="Fuchsian_equation" class="mw-redirect" title="Fuchsian equation">Fuchsian equation</a> with n singular points has a group of symmetries acting (projectively) on its solutions, isomorphic to the <a href="Coxeter_group" title="Coxeter group">Coxeter group</a> W(Dn) of order 2n−1n!. The hypergeometric equation is the case n = 3, with group of order 24 isomorphic to the symmetric group on 4 points, as first described by

<a href="Ernst_Kummer" title="Ernst Kummer">Kummer</a>. The appearance of the symmetric group is accidental and has no analogue for more than 3 singular points, and it is sometimes better to think of the group as an extension of the symmetric group on 3 points (acting as permutations of the 3 singular points) by a <a href="Klein_4-group" class="mw-redirect" title="Klein 4-group">Klein 4-group</a> (whose elements change the signs of the differences of the exponents at an even number of singular points). Kummer's group of 24 transformations is generated by the three transformations taking a solution F(a,b;c;z) to one of

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(1-z)^{-a}F\left(a,c-b;c;{\tfrac {z}{z-1}}\right)\\F(a,b;1+a+b-c;1-z)\\(1-z)^{-b}F\left(c-a,b;c;{\tfrac {z}{z-1}}\right)\end{aligned}}}</annotation>

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which correspond to the transpositions (12), (23), and (34) under an isomorphism with the symmetric group on 4 points 1, 2, 3, 4. (The first and third of these are actually equal to F(a,b;c;z) whereas the second is an independent solution to the differential equation.)

Applying Kummer's 24 = 6×4 transformations to the hypergeometric function gives the 6 = 2×3 solutions above corresponding to each of the 2 possible exponents at each of the 3 singular points, each of which appears 4 times because of the identities

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<mtext>Euler transformation</mtext>

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<mtext>Pfaff transformation</mtext>

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<mtext>Pfaff transformation</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{}_{2}F_{1}(a,b;c;z)&=(1-z)^{c-a-b}\,{}_{2}F_{1}(c-a,c-b;c;z)&&{\text{Euler transformation}}\\{}_{2}F_{1}(a,b;c;z)&=(1-z)^{-a}\,{}_{2}F_{1}(a,c-b;c;{\tfrac {z}{z-1}})&&{\text{Pfaff transformation}}\\{}_{2}F_{1}(a,b;c;z)&=(1-z)^{-b}\,{}_{2}F_{1}(c-a,b;c;{\tfrac {z}{z-1}})&&{\text{Pfaff transformation}}\end{aligned}}}</annotation>

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The hypergeometric differential equation may be brought into the Q-form

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}u}{dz^{2}}}+Q(z)u(z)=0}</annotation>

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by making the substitution u = wv and eliminating the first-derivative term. One finds that

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<annotation encoding="application/x-tex">{\displaystyle Q={\frac {z^{2}[1-(a-b)^{2}]+z[2c(a+b-1)-4ab]+c(2-c)}{4z^{2}(1-z)^{2}}}}</annotation>

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and v is given by the solution to

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<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dz}}\log v(z)=-{\frac {c-z(a+b+1)}{2z(1-z)}}=-{\frac {c}{2z}}-{\frac {1+a+b-c}{2(z-1)}}}</annotation>

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which is

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<annotation encoding="application/x-tex">{\displaystyle v(z)=z^{-c/2}(1-z)^{(c-a-b-1)/2}.}</annotation>

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The Q-form is significant in its relation to the <a href="Schwarzian_derivative" title="Schwarzian derivative">Schwarzian derivative</a> (<a href="#CITEREFHille1976">Hille 1976</a>, pp. 307–401).

<div class="mw-heading mw-heading3"><h3 id="Schwarz_triangle_maps">Schwarz triangle maps</h3></div>

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

The Schwarz triangle maps or Schwarz s-functions are ratios of pairs of solutions.

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

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where k is one of the points 0, 1, ∞. The notation

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<annotation encoding="application/x-tex">{\displaystyle D_{k}(\lambda ,\mu ,\nu ;z)=s_{k}(z)}</annotation>

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is also sometimes used. Note that the connection coefficients become <a href="M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformations</a> on the triangle maps.

Note that each triangle map is <a href="Regular_singular_point" title="Regular singular point">regular</a> at z ∈ {0, 1, ∞} respectively, with

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{0}(z)&=z^{\lambda }(1+{\mathcal {O}}(z))\\s_{1}(z)&=(1-z)^{\mu }(1+{\mathcal {O}}(1-z))\end{aligned}}}</annotation>

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and

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<annotation encoding="application/x-tex">{\displaystyle s_{\infty }(z)=z^{\nu }(1+{\mathcal {O}}({\tfrac {1}{z}})).}</annotation>

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In the special case of λ, μ and ν real, with 0 ≤ λ,μ,ν < 1 then the s-maps are <a href="Conformal_map" title="Conformal map">conformal maps</a> of the <a href="Upper_half-plane" title="Upper half-plane">upper half-plane</a> H to triangles on the <a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a>, bounded by circular arcs. This mapping is <a href="Schwarzian_derivative#Conformal_mapping_of_circular_arc_polygons" title="Schwarzian derivative">a generalization</a> of the <a href="Schwarz%E2%80%93Christoffel_mapping" title="Schwarz–Christoffel mapping">Schwarz–Christoffel mapping</a> to triangles with circular arcs. The singular points 0,1 and ∞ are sent to the triangle vertices. The angles of the triangle are πλ, πμ and πν respectively.

Furthermore, in the case of λ=1/p, μ=1/q and ν=1/r for integers p, q, r, then the triangle tiles the sphere, the complex plane or the upper half plane according to whether λ + μ + ν − 1 is positive, zero or negative; and the s-maps are inverse functions of <a href="Automorphic_function" title="Automorphic function">automorphic functions</a> for the <a href="Triangle_group" title="Triangle group">triangle group</a> 〈p, q, r〉 = Δ(p, q, r).

<div class="mw-heading mw-heading3"><h3 id="Monodromy_group">Monodromy group</h3></div>

The monodromy of a hypergeometric equation describes how fundamental solutions change when analytically continued around paths in the z plane that return to the same point.

That is, when the path winds around a singularity of 2F1, the value of the solutions at the endpoint will differ from the starting point.

Two fundamental solutions of the hypergeometric equation are related to each other by a linear transformation; thus the monodromy is a mapping (group homomorphism):

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<annotation encoding="application/x-tex">{\displaystyle \pi _{1}(\mathbf {C} \setminus \{0,1\},z_{0})\to {\text{GL}}(2,\mathbf {C} )}</annotation>

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where π1 is the <a href="Fundamental_group" title="Fundamental group">fundamental group</a>. In other words, the monodromy is a two dimensional linear representation of the fundamental group. The <a href="Monodromy_group" class="mw-redirect" title="Monodromy group">monodromy group</a> of the equation is the image of this map, i.e. the group generated by the monodromy matrices. The monodromy representation of the fundamental group can be computed explicitly in terms of the exponents at the singular points.<a href="#cite_note-3">[2]</a> If (α, α'), (β, β') and (γ,γ') are the exponents at 0, 1 and ∞, then, taking z0 near 0, the loops around 0 and 1 have monodromy matrices

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g_{0}&={\begin{pmatrix}e^{2\pi i\alpha }&0\\0&e^{2\pi i\alpha ^{\prime }}\end{pmatrix}}\\g_{1}&={\begin{pmatrix}{\mu e^{2\pi i\beta }-e^{2\pi i\beta ^{\prime }} \over \mu -1}&{\mu (e^{2\pi i\beta }-e^{2\pi i\beta ^{\prime }}) \over (\mu -1)^{2}}\\e^{2\pi i\beta ^{\prime }}-e^{2\pi i\beta }&{\mu e^{2\pi i\beta ^{\prime }}-e^{2\pi i\beta } \over \mu -1}\end{pmatrix}},\end{aligned}}}</annotation>

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where

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<annotation encoding="application/x-tex">{\displaystyle \mu ={\sin \pi (\alpha +\beta ^{\prime }+\gamma ^{\prime })\sin \pi (\alpha ^{\prime }+\beta +\gamma ^{\prime }) \over \sin \pi (\alpha ^{\prime }+\beta ^{\prime }+\gamma ^{\prime })\sin \pi (\alpha +\beta +\gamma ^{\prime })}.}</annotation>

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If 1−a, c−a−b, a−b are non-integer <a href="Rational_number" title="Rational number">rational numbers</a> with denominators k,l,m then the monodromy group is finite <a href="If_and_only_if" title="If and only if">if and only if</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/k+1/l+1/m>1}">

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<annotation encoding="application/x-tex">{\displaystyle 1/k+1/l+1/m>1}</annotation>

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</math><img src="./3e5ff7d616b6e543dc3e76432015e8cf836881c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.861ex; height:2.843ex;" alt="{\displaystyle 1/k+1/l+1/m>1}" loading="lazy">, see <a href="Schwarz's_list" title="Schwarz's list">Schwarz's list</a> or <a href="Picard%E2%80%93Vessiot_theory" title="Picard–Vessiot theory">Kovacic's algorithm</a>.

<div class="mw-heading mw-heading2"><h2 id="Integral_formulas">Integral formulas</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Euler_type">Euler type</h3></div>

If B is the <a href="Beta_function" title="Beta function">beta function</a> then

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {B} (b,c-b)\,_{2}F_{1}(a,b;c;z)=\int _{0}^{1}x^{b-1}(1-x)^{c-b-1}(1-zx)^{-a}\,dx\qquad \Re (c)>\Re (b)>0,}</annotation>

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provided that z is not a real number such that it is greater than or equal to 1. This can be proved by expanding (1 − zx)−a using the <a href="Binomial_theorem" title="Binomial theorem">binomial theorem</a> and then integrating term by term for z with absolute value smaller than 1, and by analytic continuation elsewhere. When z is a real number greater than or equal to 1, analytic continuation must be used, because (1 − zx) is zero at some point in the support of the integral, so the value of the integral may be ill-defined. This was given by Euler in 1748 and implies <a href="Binomial_transform#Euler_transform" title="Binomial transform">Euler's</a> and Pfaff's hypergeometric transformations.

Other representations, corresponding to other <a href="Principal_branch" title="Principal branch">branches</a>, are given by taking the same integrand, but taking the path of integration to be a closed <a href="Pochhammer_cycle" class="mw-redirect" title="Pochhammer cycle">Pochhammer cycle</a> enclosing the singularities in various orders. Such paths correspond to the <a href="Monodromy" title="Monodromy">monodromy</a> action.

<div class="mw-heading mw-heading3"><h3 id="Barnes_integral">Barnes integral</h3></div>

Barnes used the theory of <a href="Residue_(complex_analysis)" title="Residue (complex analysis)">residues</a> to evaluate the <a href="Barnes_integral" title="Barnes integral">Barnes integral</a>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2\pi i}}\int _{-i\infty }^{i\infty }{\frac {\Gamma (a+s)\Gamma (b+s)\Gamma (-s)}{\Gamma (c+s)}}(-z)^{s}\,ds}</annotation>

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as

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

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where the contour is drawn to separate the poles 0, 1, 2... from the poles −a, −a − 1, ..., −b, −b − 1, ... . This is valid as long as z is not a nonnegative real number.

<div class="mw-heading mw-heading3"><h3 id="John_transform">John transform</h3></div>

The Gauss hypergeometric function can be written as a <a href="John_transform" class="mw-redirect" title="John transform">John transform</a> (<a href="#CITEREFGelfandGindikinGraev2003">Gelfand, Gindikin & Graev 2003</a>, 2.1.2).

<div class="mw-heading mw-heading2"><h2 id="Gauss's_contiguous_relations">Gauss's contiguous relations</h2></div>

The six functions

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

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are called contiguous to 2F1(a, b; c; z). Gauss showed that 2F1(a, b; c; z) can be written as a linear combination of any two of its contiguous functions, with rational coefficients in terms of a, b, c, and z. This gives

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<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}6\\2\end{pmatrix}}=15}</annotation>

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relations, given by identifying any two lines on the right hand side of

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}z{\frac {dF}{dz}}&=z{\frac {ab}{c}}F(a+,b+,c+)\\&=a(F(a+)-F)\\&=b(F(b+)-F)\\&=(c-1)(F(c-)-F)\\&={\frac {(c-a)F(a-)+(a-c+bz)F}{1-z}}\\&={\frac {(c-b)F(b-)+(b-c+az)F}{1-z}}\\&=z{\frac {(c-a)(c-b)F(c+)+c(a+b-c)F}{c(1-z)}}\end{aligned}}}</annotation>

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where F = 2F1(a, b; c; z), F(a+) = 2F1(a + 1, b; c; z), and so on. Repeatedly applying these relations gives a linear relation over C(z) between any three functions of the form

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

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where m, n, and l are integers.<a href="#cite_note-4">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="Gauss's_continued_fraction">Gauss's continued fraction</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Gauss_continued_fraction" class="mw-redirect" title="Gauss continued fraction">Gauss continued fraction</a></div>

Gauss used the contiguous relations to give several ways to write a quotient of two hypergeometric functions as a continued fraction, for example:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {{}_{2}F_{1}(a+1,b;c+1;z)}{{}_{2}F_{1}(a,b;c;z)}}={\cfrac {1}{1+{\cfrac {{\frac {(a-c)b}{c(c+1)}}z}{1+{\cfrac {{\frac {(b-c-1)(a+1)}{(c+1)(c+2)}}z}{1+{\cfrac {{\frac {(a-c-1)(b+1)}{(c+2)(c+3)}}z}{1+{\cfrac {{\frac {(b-c-2)(a+2)}{(c+3)(c+4)}}z}{1+{}\ddots }}}}}}}}}}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Transformation_formulas">Transformation formulas</h2></div>

Transformation formulas relate two hypergeometric functions at different values of the argument z.

<div class="mw-heading mw-heading3"><h3 id="Fractional_linear_transformations">Fractional linear transformations</h3></div>

Euler's transformation is

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

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It follows by combining the two Pfaff transformations

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{}_{2}F_{1}(a,b;c;z)&=(1-z)^{-b}{}_{2}F_{1}\left(b,c-a;c;{\tfrac {z}{z-1}}\right)\\{}_{2}F_{1}(a,b;c;z)&=(1-z)^{-a}{}_{2}F_{1}\left(a,c-b;c;{\tfrac {z}{z-1}}\right)\\\end{aligned}}}</annotation>

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which in turn follow from Euler's integral representation. For extension of Euler's first and second transformations, see <a href="#CITEREFRathieParis2007">Rathie & Paris (2007)</a> and <a href="#CITEREFRakhaRathie2011">Rakha & Rathie (2011)</a>.

It can also be written as linear combination

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{}_{2}F_{1}(a,b;c,z)={}&{\frac {\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}}{}_{2}F_{1}(a,b;a+b+1-c;1-z)\\[6pt]&{}+{\frac {\Gamma (c)\Gamma (a+b-c)}{\Gamma (a)\Gamma (b)}}(1-z)^{c-a-b}{}_{2}F_{1}(c-a,c-b;1+c-a-b;1-z).\end{aligned}}}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Quadratic_transformations">Quadratic transformations</h3></div>

If two of the numbers 1 − c, c − 1, a − b, b − a, a + b − c, c − a − b are equal or one of them is 1/2 then there is a quadratic transformation of the hypergeometric function, connecting it to a different value of z related by a quadratic equation. The first examples were given by <a href="#CITEREFKummer1836">Kummer (1836)</a>, and a complete list was given by <a href="#CITEREFGoursat1881">Goursat (1881)</a>. A typical example is

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

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<div class="mw-heading mw-heading3"><h3 id="Higher_order_transformations">Higher order transformations</h3></div>

If 1−c, a−b, a+b−c differ by signs or two of them are 1/3 or −1/3 then there is a cubic transformation of the hypergeometric function, connecting it to a different value of z related by a cubic equation. The first examples were given by <a href="#CITEREFGoursat1881">Goursat (1881)</a>. A typical example is

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

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There are also some transformations of degree 4 and 6. Transformations of other degrees only exist if a, b, and c are certain rational numbers (<a href="#CITEREFVidunas2005">Vidunas 2005</a>). For example,

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<annotation encoding="application/x-tex">{\displaystyle {}_{2}F_{1}\left({\tfrac {1}{4}},{\tfrac {3}{8}};{\tfrac {7}{8}};z\right)(z^{4}-60z^{3}+134z^{2}-60z+1)^{1/16}={}_{2}F_{1}\left({\tfrac {1}{48}},{\tfrac {17}{48}};{\tfrac {7}{8}};{\tfrac {-432z(z-1)^{2}(z+1)^{8}}{(z^{4}-60z^{3}+134z^{2}-60z+1)^{3}}}\right).}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Values_at_special_points_z">Values at special points z</h2></div>

See <a href="#CITEREFSlater1966">Slater (1966</a>, Appendix III) for a list of summation formulas at special points, most of which also appear in <a href="#CITEREFBailey1935">Bailey (1935)</a>. <a href="#CITEREFGesselStanton1982">Gessel & Stanton (1982)</a> gives further evaluations at more points. <a href="#CITEREFKoepf1995">Koepf (1995)</a> shows how most of these identities can be verified by computer algorithms.

<div class="mw-heading mw-heading3"><h3 id="Special_values_at_z_=_1">Special values at z = 1</h3></div>

Gauss's summation theorem, named for <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>, is the identity

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

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which follows from Euler's integral formula by putting z = 1. It includes the <a href="Vandermonde_identity" class="mw-redirect" title="Vandermonde identity">Vandermonde identity</a> as a special case.

For the special case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-m}">

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

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<a href="Bilateral_hypergeometric_series" title="Bilateral hypergeometric series">Dougall's formula</a> generalizes this to the <a href="Bilateral_hypergeometric_series" title="Bilateral hypergeometric series">bilateral hypergeometric series</a> at z = 1.

<div class="mw-heading mw-heading3"><h3 id="Kummer's_theorem_(z_=_−1)">Kummer's theorem (z = −1)</h3></div>

There are many cases where hypergeometric functions can be evaluated at z = −1 by using a quadratic transformation to change z = −1 to z = 1 and then using Gauss's theorem to evaluate the result. A typical example is Kummer's theorem, named for <a href="Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a>:

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

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which follows from Kummer's quadratic transformations

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}_{2}F_{1}(a,b;1+a-b;z)&=(1-z)^{-a}\;_{2}F_{1}\left({\frac {a}{2}},{\frac {1+a}{2}}-b;1+a-b;-{\frac {4z}{(1-z)^{2}}}\right)\\&=(1+z)^{-a}\,_{2}F_{1}\left({\frac {a}{2}},{\frac {a+1}{2}};1+a-b;{\frac {4z}{(1+z)^{2}}}\right)\end{aligned}}}</annotation>

</semantics>

</math>

and Gauss's theorem by putting z = −1 in the first identity. For generalization of Kummer's summation, see <a href="#CITEREFLavoieGrondinRathie1996">Lavoie, Grondin & Rathie (1996)</a>.

<div class="mw-heading mw-heading3"><h3 id="Values_at_z_=_1/2">Values at z = 1/2</h3></div>

Gauss's second summation theorem is

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

</semantics>

</math>

Bailey's theorem is

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

</semantics>

</math>

For generalizations of Gauss's second summation theorem and Bailey's summation theorem, see <a href="#CITEREFLavoieGrondinRathie1996">Lavoie, Grondin & Rathie (1996)</a>.

<div class="mw-heading mw-heading3"><h3 id="Other_points">Other points</h3></div>

There are many other formulas giving the hypergeometric function as an algebraic number at special rational values of the parameters, some of which are listed in <a href="#CITEREFGesselStanton1982">Gessel & Stanton (1982)</a> and <a href="#CITEREFKoepf1995">Koepf (1995)</a>. Some typical examples are given by

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

</semantics>

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which can be restated as

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<annotation encoding="application/x-tex">{\displaystyle T_{a}(\cos x)={}_{2}F_{1}\left(a,-a;{\tfrac {1}{2}};{\tfrac {1}{2}}(1-\cos x)\right)=\cos(ax)}</annotation>

</semantics>

</math>

whenever −π < x < π and T is the (generalized) <a href="Chebyshev_polynomial" class="mw-redirect" title="Chebyshev polynomial">Chebyshev polynomial</a>.

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

<li><a href="Basic_hypergeometric_series" title="Basic hypergeometric series">Basic hypergeometric series</a></li>

<li><a href="Bilateral_hypergeometric_series" title="Bilateral hypergeometric series">Bilateral hypergeometric series</a></li>

<li><a href="Elliptic_hypergeometric_series" title="Elliptic hypergeometric series">Elliptic hypergeometric series</a></li>

<li><a href="General_hypergeometric_function" title="General hypergeometric function">General hypergeometric function</a></li>

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

<li><a href="Hypergeometric_distribution" title="Hypergeometric distribution">Hypergeometric distribution</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></ul>

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

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFMorita1996" class="citation journal cs1">Morita, Tohru (1996). "Use of the Gauss contiguous relations in computing the hypergeometric functions F(n+1/2,n+1/2;m;z)". Interd. Inf. Sci. 2 (1): 63–74. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4036%2Fiis.1996.63">10.4036/iis.1996.63</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1398101">1398101</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFInce1944">Ince 1944</a>, pp. 393–393

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFRakhaRathieChopra2011" class="citation journal cs1">Rakha, Medhat A.; Rathie, Arjun K.; Chopra, Purnima (2011). "On some new contiguous relations for the Gauss hypergeometric function with applications". Comput. Math. Appl. 61 (3): 620–629. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.camwa.2010.12.008">10.1016/j.camwa.2010.12.008</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2764057">2764057</a>.</cite>

</li>

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

<li id="cite_note-1"><a href="#cite_ref-1">^</a> This convention is common in hypergeometric function theory, but it is the opposite convention to the one used in <a href="Falling_and_rising_factorials" title="Falling and rising factorials">Falling and rising factorials</a>.

</li>

</ol></div><div class="reflist reflist-lower-alpha">

</div>

<ul><li><cite id="CITEREFAndrewsAskeyRoy1999" class="citation book cs1"><a href="George_Andrews_(mathematician)" title="George Andrews (mathematician)">Andrews, George E.</a>; 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>978-0-521-62321-6</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). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170624164738/http://plouffe.fr/simon/math/Bailey%20W.N.%20Generalized%20Hypergeometric%20Series%20(1964)(L)(T)(59s).pdf">Generalized Hypergeometric Series</a> (PDF). Cambridge University Press. Archived from <a rel="nofollow" class="external text" href="http://plouffe.fr/simon/math/Bailey%20W.N.%20Generalized%20Hypergeometric%20Series%20%281964%29%28L%29%28T%29%2859s%29.pdf">the original</a> (PDF) on 2017-06-24. Retrieved 2016-07-23.</cite></li>

<li><a href="Frits_Beukers" title="Frits Beukers">Beukers, Frits</a> (2002), <a rel="nofollow" class="external text" href="https://webspace.science.uu.nl/~beuke106/MRIcourse93.ps">Gauss' hypergeometric function</a>. (lecture notes reviewing basics, as well as triangle maps and monodromy)</li>

<li><cite id="CITEREFOlde_Daalhuis2010" class="citation cs2">Olde Daalhuis, Adri B. (2010), <a rel="nofollow" class="external text" href="http://dlmf.nist.gov/15">"Hypergeometric function"</a>, in <a href="Frank_W._J._Olver" title="Frank W. J. Olver">Olver, Frank W. J.</a>; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), <a href="Digital_Library_of_Mathematical_Functions" title="Digital Library of Mathematical Functions">NIST Handbook of Mathematical Functions</a>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-19225-5</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=2723248">2723248</a></cite>.</li>

<li><cite id="CITEREFErdélyiMagnusOberhettingerTricomi1953" class="citation book cs1"><a href="Arthur_Erd%C3%A9lyi" title="Arthur Erdélyi">Erdélyi, Arthur</a>; <a href="Wilhelm_Magnus" title="Wilhelm Magnus">Magnus, Wilhelm</a>; Oberhettinger, Fritz & Tricomi, Francesco G. (1953). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110811153220/http://apps.nrbook.com/bateman/Vol1.pdf">Higher transcendental functions</a> (PDF). Vol. I. New York – Toronto – London: McGraw–Hill Book Company, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-89874-206-0</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=0058756">0058756</a>. Archived from <a rel="nofollow" class="external text" href="http://apps.nrbook.com/bateman/Vol1.pdf">the original</a> (PDF) on 2011-08-11. Retrieved 2011-07-30.</cite> <code class="cs1-code">{{cite book}}</code>: ISBN / Date incompatibility (help)</li>

<li><a href="George_Gasper" title="George Gasper">Gasper, George</a> & <a href="Mizan_Rahman" title="Mizan Rahman">Rahman, Mizan</a> (2004). Basic Hypergeometric Series, 2nd Edition, Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-83357-4</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 seriem 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.}}}">

<mfrac>

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</mrow>

<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></li>

<li><cite id="CITEREFGelfandGindikinGraev2003" class="citation book cs1"><a href="Israel_Gelfand" title="Israel Gelfand">Gelfand, I. M.</a>; <a href="Simon_Gindikin" title="Simon Gindikin">Gindikin, S.G.</a> & <a href="Mark_Iosifovich_Graev" title="Mark Iosifovich Graev">Graev, M.I.</a> (2003) [2000]. <a rel="nofollow" class="external text" href="https://books.google.com/books?isbn=0821829327">Selected topics in integral geometry</a>. Translations of Mathematical Monographs. Vol. 220. Providence, R.I.: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-2932-5</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=2000133">2000133</a>.</cite></li>

<li><cite id="CITEREFGesselStanton1982" class="citation journal cs1"><a href="Ira_Gessel" title="Ira Gessel">Gessel, Ira</a> & Stanton, Dennis (1982). "Strange evaluations of hypergeometric series". SIAM Journal on Mathematical Analysis. 13 (2): 295–308. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0513021">10.1137/0513021</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0036-1410">0036-1410</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0647127">0647127</a>.</cite></li>

<li><cite id="CITEREFGoursat1881" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="%C3%89douard_Goursat" title="Édouard Goursat">Goursat, Édouard</a> (1881). <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=ASENS_1881_2_10__S3_0">"Sur l'équation différentielle linéaire, qui admet pour intégrale la série hypergéométrique"</a>. Annales Scientifiques de l'École Normale Supérieure (in French). 10: 3–142. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fasens.207">10.24033/asens.207</a>. Retrieved 2008-10-16.</cite></li>

<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>0-12-336170-2</bdi>.</cite> (part 1 treats hypergeometric functions on Lie groups)</li>

<li><cite id="CITEREFHille1976" class="citation book cs1"><a href="Einar_Hille" title="Einar Hille">Hille, Einar</a> (1976). <a rel="nofollow" class="external text" href="https://archive.org/details/ordinarydifferen00hill_0">Ordinary differential equations in the complex domain</a>. Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-69620-0</bdi>.</cite></li>

<li><cite id="CITEREFInce1944" class="citation book cs1"><a href="E._L._Ince" class="mw-redirect" title="E. L. Ince">Ince, E. L.</a> (1944). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.476224">Ordinary Differential Equations</a>. Dover Publications.</cite></li>

<li><cite id="CITEREFKlein1981" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="Felix_Klein" title="Felix Klein">Klein, Felix</a> (1981). <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?PPN375394591">Vorlesungen über die hypergeometrische Funktion</a>. Grundlehren der Mathematischen Wissenschaften (in German). Vol. 39. Berlin, New York: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-10455-1</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=0668700">0668700</a>.</cite></li>

<li><cite id="CITEREFKoepf1995" class="citation journal cs1">Koepf, Wolfram (1995). <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjsco.1995.1056">"Algorithms for m-fold hypergeometric summation"</a>. Journal of Symbolic Computation. 20 (4): 399–417. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjsco.1995.1056">10.1006/jsco.1995.1056</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0747-7171">0747-7171</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1384455">1384455</a>.</cite></li>

<li><cite id="CITEREFKummer1836" class="citation journal cs1 cs1-prop-foreign-lang-source">Kummer, Ernst Eduard (1836). <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?GDZPPN00214056X">"Über die hypergeometrische Reihe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\tfrac {\alpha \cdot \beta }{1\cdot \gamma }}~x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}x^{2}+{\tfrac {\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1\cdot 2\cdot 3\cdot \gamma (\gamma +1)(\gamma +2)}}x^{3}+\cdots }">

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<annotation encoding="application/x-tex">{\displaystyle 1+{\tfrac {\alpha \cdot \beta }{1\cdot \gamma }}~x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}x^{2}+{\tfrac {\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1\cdot 2\cdot 3\cdot \gamma (\gamma +1)(\gamma +2)}}x^{3}+\cdots }</annotation>

</semantics>

</math><img src="./a24bc1c17525b5489ed65732c5ccc99366ec340e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:57.133ex; height:4.843ex;" alt="{\displaystyle 1+{\tfrac {\alpha \cdot \beta }{1\cdot \gamma }}~x+{\tfrac {\alpha (\alpha +1)\beta (\beta +1)}{1\cdot 2\cdot \gamma (\gamma +1)}}x^{2}+{\tfrac {\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{1\cdot 2\cdot 3\cdot \gamma (\gamma +1)(\gamma +2)}}x^{3}+\cdots }" loading="lazy">"</a>. <a href="Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" class="mw-redirect" title="Journal für die reine und angewandte Mathematik">Journal für die reine und angewandte Mathematik</a> (in German). 15: 39–83, 127–172. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0075-4102">0075-4102</a>.</cite></li>

<li><cite id="CITEREFLavoieGrondinRathie1996" class="citation journal cs1">Lavoie, J. L.; Grondin, F.; Rathie, A.K. (1996). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0377-0427%2895%2900279-0">"Generalizations of Whipple's theorem on the sum of a 3F2"</a>. J. Comput. Appl. Math. 72 (2): 293–300. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0377-0427%2895%2900279-0">10.1016/0377-0427(95)00279-0</a>.</cite></li>

<li><cite id="CITEREFPressTeukolskyVetterlingFlannery2007" class="citation book cs1">Press, W.H.; Teukolsky, S.A.; Vetterling, W.T. & Flannery, B.P. (2007). <a rel="nofollow" class="external text" href="http://apps.nrbook.com/empanel/index.html#pg=318">"Section 6.13. Hypergeometric Functions"</a>. Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-88068-8</bdi>.</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="CITEREFRathieParis2007" class="citation journal cs1">Rathie, Arjun K.; Paris, R.B. (2007). "An extension of the Euler's-type transformation for the 3F2 series". Far East J. Math. Sci. 27 (1): 43–48.</cite></li>

<li><cite id="CITEREFRiemann1857" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Bernhard_Riemann" title="Bernhard Riemann">Riemann, Bernhard</a> (1857). <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002018691">"Beiträge zur Theorie der durch die Gauss'sche Reihe F(α, β, γ, x) darstellbaren Functionen"</a>. Abhandlungen der Mathematischen Classe der Königlichen Gesellschaft der Wissenschaften zu Göttingen (in German). 7. Göttingen: Verlag der Dieterichschen Buchhandlung: 3–22.</cite> (a reprint of this paper can be found in <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.emis.de/classics/Riemann/PFunct.pdf">"All publications of Riemann"</a> (PDF).</cite>)</li>

<li><cite id="CITEREFSlater1960" class="citation book cs1"><a href="Lucy_Joan_Slater" title="Lucy Joan Slater">Slater, Lucy Joan</a> (1960). <a rel="nofollow" class="external text" href="https://archive.org/details/confluenthyperge0000slat">Confluent hypergeometric functions</a>. Cambridge, UK: Cambridge University Press. <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=0107026">0107026</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>0-521-06483-X</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>.</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="CITEREFVidunas2005" class="citation journal cs1">Vidunas, Raimundas (2005). "Transformations of some Gauss hypergeometric functions". Journal of Symbolic Computation. 178 (1–2): 473–487. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0310436">math/0310436</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/2005JCoAM.178..473V">2005JCoAM.178..473V</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cam.2004.09.053">10.1016/j.cam.2004.09.053</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:119596800">119596800</a>.</cite></li>

<li><cite id="CITEREFWall1948" class="citation book cs1">Wall, H.S. (1948). Analytic Theory of Continued Fractions. D. Van Nostrand Company, Inc.</cite></li>

<li><cite id="CITEREFWhittakerWatson1927" class="citation book cs1"><a href="E._T._Whittaker" title="E. T. Whittaker">Whittaker, E.T.</a> & <a href="G._N._Watson" title="G. N. Watson">Watson, G.N.</a> (1927). <a href="A_Course_of_Modern_Analysis" title="A Course of Modern Analysis">A Course of Modern Analysis</a>. Cambridge, UK: Cambridge University Press.</cite></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>3-528-06925-2</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=1453580">1453580</a>.</cite></li></ul>

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

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

<li>John Pearson, <a rel="nofollow" class="external text" href="http://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf">Computation of Hypergeometric Functions</a> (<a href="University_of_Oxford" title="University of Oxford">University of Oxford</a>, MSc Thesis)</li>

<li>Marko Petkovsek, Herbert Wilf and Doron Zeilberger, <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> (freely downloadable)</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></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%"></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>

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