# Hypergeometric function

A solution of a hypergeometric equation

$$\tag{1 } z ( 1 - z) w ^ {\prime\prime} + [ \gamma - ( \alpha + \beta + 1) z] w ^ \prime - \alpha \beta w = 0.$$

A hypergeometric function can be defined with the aid of the so-called Gauss series

$$\tag{2 } F ( \alpha , \beta ; \gamma ; z) = \ {} _ {2} F _ {1} ( \alpha , \beta ; \gamma ; z) = \ F ( \beta , \alpha ; \gamma ; z) =$$

$$= \ \sum _ {n = 0 } ^ \infty \frac{( \alpha ) _ {n} ( \beta ) _ {n} }{( \gamma ) _ {n} } \frac{z ^ {n} }{n!\ } =$$

$$= \ \frac{\Gamma ( \gamma ) }{\Gamma ( \alpha ) \Gamma ( \beta ) } \sum _ {n = 0 } ^ \infty \frac{\Gamma ( \alpha + n) \Gamma ( \beta + n) }{\Gamma ( \gamma + n) } \frac{z ^ {n} }{n!} ,$$

where $\alpha , \beta , \gamma$ are parameters which assume arbitrary real or complex values except for $\gamma = 0, - 1, - 2 ,\dots$; $z$ is a complex variable; and $( x) _ {n} \equiv x( x + 1) \dots ( x + n - 1)$. The function $F( \alpha , \beta ; \gamma ; z)$ is called a hypergeometric function of the first kind. The second linearly independent solution of (1),

$$\Phi ( \alpha , \beta ; \gamma ; z) =$$

$$= \ \frac{\Gamma ( \alpha - \gamma + 1) \Gamma ( \beta - \gamma + 1) }{\Gamma ( \alpha ) \Gamma ( \beta ) \Gamma ( 1 - \gamma ) } z ^ {1 - \gamma } F ( \alpha - \gamma + 1, \beta - \gamma + 1; 2 - \gamma ; z),$$

is called a hypergeometric function of the second kind.

The series (2) is absolutely and uniformly convergent if $| z | < 1$; the convergence also extends over the unit circle if $\mathop{\rm Re} ( \alpha + \beta - \gamma ) < 0$; if $0 \leq \mathop{\rm Re} ( \alpha + \beta - \gamma ) < 1$ it converges at all points of the unit circle except $z = 1$. However, there exists an analytic continuation of the hypergeometric function (2) to the exterior $| z | > 1$ of the unit disc with the slit $( 1, \infty )$. The function $F ( \alpha , \beta ; \gamma ; z )$ is a univalent analytic function in the complex $z$-plane with slit $( 1, \infty )$. If $\alpha$ or $\beta$ are zero or negative integers, the series (2) terminates after a finite number of terms, and the hypergeometric function is a polynomial in $z$.

If $\gamma = - n$, $n = 0, 1, \dots$ the hypergeometric function is not defined, but

$$\lim\limits _ {\gamma \rightarrow - n } \ \frac{F ( \alpha , \beta ; \gamma ; z) }{F ( \gamma ) } =$$

$$= \ \frac{( \alpha ) _ {n + 1 } ( \beta ) _ {n + 1 } }{( n + 1)! } z ^ {n + 1 } F ( \alpha + n + 1, \beta + n + 1; n + 2; z).$$

Elementary relations. The six functions

$$F ( \alpha \pm 1, \beta ; \gamma ; z),\ \ F ( \alpha , \beta \pm 1; \gamma ; z) \ \ F ( \alpha , \beta ; \gamma \pm 1; z)$$

are said to be contiguous to the hypergeometric function $F ( \alpha , \beta ; \gamma ; z )$. There exists a linear relationship between that function and any two functions which are contiguous to it. For instance, C.F. Gauss ,  was the first to find 15 formulas of the type

$$\gamma F ( \alpha , \beta - 1; \gamma ; z) + ( \alpha - \beta ) zF ( \alpha , \beta ; \gamma + 1; z) =$$

$$= \ \gamma F ( \alpha - 1, \beta ; \gamma ; z).$$

The associated functions $F ( \alpha + m , \beta + n ; \gamma + l ; z)$, where $m, n, l$ are integers, can be obtained by iterated application of Gauss' relations. The following differentiation formulas apply:

$$\frac{d ^ {n} }{dz ^ {n} } F ( \alpha , \beta ; \gamma ; z) = \ \frac{( \alpha ) _ {n} ( \beta ) _ {n} }{( \gamma ) _ {n} } F ( \alpha + n, \beta + n; \gamma + n; z).$$

Equation (1) has 24 solutions of the form

$$z ^ \rho ( 1 - z) ^ \sigma F ( \alpha ^ \prime , \beta ^ \prime ; \gamma ^ \prime ; z ^ \prime ),$$

where $\rho$, $\sigma$, $\alpha ^ \prime$, $\beta ^ \prime$, and $\gamma ^ \prime$ are linear functions of $\alpha$, $\beta$ and $\gamma$; and $z$ and $z ^ \prime$ are connected by a bilinear transformation. Any three solutions are linearly dependent . There exist square, cubic and higher-order transformations .

Principal integral representations. If $\mathop{\rm Re} \gamma > \mathop{\rm Re} \beta > 0$ and $| \mathop{\rm arg} ( 1 - x) | < \pi$, Euler's formula

$$\tag{3 } F ( \alpha , \beta ; \gamma ; z) =$$

$$= \ \frac{\Gamma ( \gamma ) }{\Gamma ( \beta ) \Gamma ( \gamma - \beta ) } \int\limits _ { 0 } ^ { 1 } t ^ {\beta - 1 } ( 1 - t) ^ {\gamma - \beta - 1 } ( 1 - tz) ^ {- \alpha } dt$$

holds. By expanding $( 1 - tz) ^ {- \alpha }$ into a binomial series and using contour integrals for the beta-function, other integral representations can be obtained . The integral (3) and other similar formulas defining an analytic function of $z$ which is single-valued throughout the $z$-plane can also be used as analytic continuations of $F ( \alpha , \beta ; \gamma ; z)$ into the domain $| \mathop{\rm arg} (- z) | < \pi$. Other analytic continuations also exist , .

The asymptotic behaviour of hypergeometric functions for large values of $| z |$ is completely described by formulas yielding analytic continuations in a neighbourhood of the point $z = \infty$ , , . If $\alpha$, $\beta$ and $z$ are given and $| \gamma |$ is sufficiently large, $| \mathop{\rm arg} \gamma | < ( \pi - \epsilon )$, $\epsilon > 0$, then, if $| z | < 1$:

$$F ( \alpha , \beta ; \gamma ; z) = \ \sum _ {k = 0 } ^ { n } \frac{( \alpha ) _ {k} ( \beta ) _ {k} }{( \gamma ) _ {k} } \frac{z ^ {k} }{k!} + O ( | \gamma | ^ {- n + 1 } ).$$

A similar expression is obtained for $| z | > 1$.

For fixed $\alpha$, $\gamma$ and $z$, $\gamma \neq 0, - 1, - 2 \dots$ $0 < | z | < 1$, and $\beta \rightarrow \infty$, $- 3 \pi /2 < \mathop{\rm arg} \beta z < \pi /2$,

$$F ( \alpha , \beta ; \gamma ; z) = \ F \left ( \alpha , \beta ; \gamma ; \ \frac{\beta z } \beta \right ) =$$

$$= \ \left [ \sum _ {n = 0 } ^ \infty \frac{( \alpha ) _ {n} ( \beta ) _ {n} }{( \gamma ) _ {n} n! } \right ] [ 1 + O ( | \beta | ^ {- 1} )].$$

Representation of functions by hypergeometric functions. The elementary functions:

$$( 1 + z) ^ {n} = \ F (- n, 1; 1; - z),$$

$$\mathop{\rm ln} \frac{1 + z }{1 - z } = 2zF \left ( { \frac{1}{2} } , 1; { \frac{3}{2} } ; z ^ {2} \right ) ,$$

$$\mathop{\rm ln} ( 1 + z) = zF ( 1, 1; 2; - z),$$

$$e ^ {z} = \lim\limits _ {b \rightarrow \infty } F \left ( 1, b; 1; { \frac{z}{b} } \right ) ,$$

$$\mathop{\rm arc} \sin z = zF \left ( { \frac{1}{2} } ,\ { \frac{1}{2} } ; { \frac{3}{2} } ; z ^ {2} \right ) ,$$

$$\mathop{\rm arc} \mathop{\rm tan} z = zF \left ( { \frac{1}{2} } , 1; { \frac{3}{2} } ; - z ^ {2} \right ) ,$$

$$\sin nz = n \sin z F \left ( \frac{1 + n }{2} , { \frac{1 - n }{2} } ; { \frac{3}{2} } ; \sin ^ {2} z \right ) ,$$

$$\cos nz = F \left ( { \frac{1}{2} } n, - { \frac{1}{2} } n; { \frac{1}{2} } ; \sin ^ {2} z \right ) .$$

The complete elliptic integrals of the first and second kinds (cf. Elliptic integral):

$$K ( z) = { \frac \pi {2} } F \left ( { \frac{1}{2} } , { \frac{1}{2} } ; 1; z ^ {2} \right ) ,$$

$$E ( z) = { \frac \pi {2} } F \left ( - { \frac{1}{2} } , { \frac{1}{2} } ; 1; z ^ {2} \right ) .$$

$$P _ {n} ^ {m} ( z) = \ \frac{( z + 1) ^ {m / 2 } }{( z - 1) ^ {m / 2 } } { \frac{1}{\Gamma ( 1 - m) } } F \left ( - n, n + 1; 1 - m; { \frac{1 - z }{2} } \right ) .$$

$$T _ {n} ( z) = F \left ( - n, n; { \frac{1}{2} } ; { \frac{1 - z }{2} } \right ) .$$

$$P _ {n} ( z) = F \left ( - n, n + 1; 1; { \frac{1 - z }{2} } \right ) .$$

$$\frac{n!}{{( 2a)} _ {n} } C _ {n} ^ {( a)} ( z) = F \left ( - n, n + 2a; a + { \frac{1}{2} } ; { \frac{1 - z }{2} } \right ) .$$

$$\frac{n! }{( a + 1) _ {n} } P _ {n} ^ {( a, b) } ( z) = F \left ( - n, a + 1 + b + n; a + 1; { \frac{1 - z }{2} } \right ) .$$

Generalizations of hypergeometric functions. The generalized hypergeometric function

$${} _ {p} F _ {q} ( \alpha _ {1} \dots \alpha _ {p} ; \ \gamma _ {1} \dots \gamma _ {q} ; z) = \ \sum _ {n = 0 } ^ \infty { \frac{1}{n!} } \frac{( \alpha _ {1} ) _ {n} \dots ( \alpha _ {p} ) _ {n} }{( \gamma _ {1} ) _ {n} \dots ( \gamma _ {q} ) _ {n} } z ^ {n}$$

is said to be the solution of the hypergeometric equation of order $q + 1$ . There are also other generalizations of hypergeometric functions, such as generalizations to include the case of several variables .

How to Cite This Entry:
Hypergeometric function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hypergeometric_function&oldid=51949
This article was adapted from an original article by E.A. Chistova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article