Hypergeometric function

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A solution of a hypergeometric equation


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


where are parameters which assume arbitrary real or complex values except for ; is a complex variable; and . The function is called a hypergeometric function of the first kind. The second linearly independent solution of (1),

is called a hypergeometric function of the second kind.

The series (2) is absolutely and uniformly convergent if ; the convergence also extends over the unit circle if ; if it converges at all points of the unit circle except . However, there exists an analytic continuation of the hypergeometric function (2) to the exterior of the unit disc with the slit [1]. The function is a univalent analytic function in the complex -plane with slit . If or are zero or negative integers, the series (2) terminates after a finite number of terms, and the hypergeometric function is a polynomial in .

If , the hypergeometric function is not defined, but

Elementary relations. The six functions

are said to be contiguous to the hypergeometric function . There exists a linear relationship between that function and any two functions which are contiguous to it. For instance, C.F. Gauss [2], [3] was the first to find 15 formulas of the type

The associated functions , where are integers, can be obtained by iterated application of Gauss' relations. The following differentiation formulas apply:

Equation (1) has 24 solutions of the form

where , , , , and are linear functions of , and ; and and are connected by a bilinear transformation. Any three solutions are linearly dependent [2]. There exist square, cubic and higher-order transformations [2][5].

Principal integral representations. If and , Euler's formula


holds. By expanding into a binomial series and using contour integrals for the beta-function, other integral representations can be obtained [2]. The integral (3) and other similar formulas defining an analytic function of which is single-valued throughout the -plane can also be used as analytic continuations of into the domain . Other analytic continuations also exist [1], [2].

The asymptotic behaviour of hypergeometric functions for large values of is completely described by formulas yielding analytic continuations in a neighbourhood of the point [1], [2], [3]. If , and are given and is sufficiently large, , , then, if :

A similar expression is obtained for .

For fixed , and , , and , ,

See also [2], [5], [6].

Representation of functions by hypergeometric functions. The elementary functions:

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

The adjoint Legendre functions:

The Chebyshev polynomials:

The Legendre polynomials:

The ultraspherical polynomials:

The Jacobi polynomials:

Generalizations of hypergeometric functions. The generalized hypergeometric function

is said to be the solution of the hypergeometric equation of order [2]. There are also other generalizations of hypergeometric functions, such as generalizations to include the case of several variables [2].


[1] N.N. Lebedev, "Special functions and their applications" , Prentice-Hall (1965) (Translated from Russian)
[2] H. Bateman (ed.) A. Erdélyi (ed.) , Higher transcendental functions , 1. The gamma function. The hypergeometric functions. Legendre functions , McGraw-Hill (1953)
[3] I.S. Gradshtein, I.M. Ryzhik, "Table of integrals, series and products" , Acad. Press (1980) (Translated from Russian)
[4] E.E. Kummer, "Ueber die hypergeometrische Reihe " J. Reine Angew. Math. , 15 (1836) pp. 39–83; 127–172
[5] A. Segun, M. Abramowitz, "Handbook of mathematical functions" , Appl. Math. Ser. , 55 , Nat. Bur. Standards (1970)
[6] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1952)
[7] A.L. Lebedev, R.M. Fedorova, "Handbook of mathematical tables" , Moscow (1956) (In Russian)
[8] N.M. Burunova, "Handbook of mathematical tables" , Moscow (1959) (In Russian) (Supplement I)
[9] A.A. Fletcher, J.C.P. Miller, L. Rosenhead, L.J. Comrie, "An index of mathematical tables" , 1–2 , Oxford Univ. Press (1962)


To the list of functions representable by hypergeometric functions the Jacobi functions should be added:

cf. [a2].

An important generalization is given by the basic hypergeometric functions, cf. [a1].


[a1] G. Gasper, M. Rahman, "Basic hypergeometric series" , Cambridge Univ. Press (1989)
[a2] T.H. Koornwinder, "Jacobi functions and analysis on noncompact semisimple Lie groups" R.A. Askey (ed.) T.H. Koornwinder (ed.) W. Schempp (ed.) , Special functions: group theoretical aspects and applications , Reidel (1984) pp. 1–85
How to Cite This Entry:
Hypergeometric function. E.A. Chistova (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098