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 . 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 ,  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 . There exist square, cubic and higher-order transformations –.
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 . 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 , .
The asymptotic behaviour of hypergeometric functions for large values of is completely described by formulas yielding analytic continuations in a neighbourhood of the point , , . If , and are given and is sufficiently large, , , then, if :
A similar expression is obtained for .
For fixed , and , , and , ,
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 Legendre polynomials:
The Jacobi polynomials:
Generalizations of hypergeometric functions. The generalized hypergeometric function
is said to be the solution of the hypergeometric equation of order . There are also other generalizations of hypergeometric functions, such as generalizations to include the case of several variables .
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To the list of functions representable by hypergeometric functions the Jacobi functions should be added:
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|
Hypergeometric function. E.A. Chistova (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hypergeometric_function&oldid=12873