# Hypergeometric Functions

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Hypergeometric Functions

analytic functions defined for ǀzǀ< 1 using hypergeometric series. The name “hypergeometric function” was coined by J. Wallis in 1650. Hypergeometric functions are integrals of the hypergeometric equation

z(1 - z)w” + [γ - (1 + α + β)Z]w´ - αβw = 0

This equation has three regular singular points 0, 1, and », and is the canonical form of hypergeometric-type equations. The most important functions of mathematical analysis are integrals of equations of the hypergeometric type (for example, spherical functions) or of equations resulting from the hypergeometric-type equations by merging their singular points (for example, cylindrical functions). The theory of hypergeometric-type equations became the basis for the origin of an important mathematical discipline, the analytic theory of differential equations.

Between various hypergeometric functions

w = F(α, β γ; z)

there are numerous relationships, for example,

F(α, 1; γ, z) = (1 - z)-1F(1, γ - α;γ; z/(z - 1))

### REFERENCE

Whittaker, E. T., and G. N. Watson. Kurs sovremennogo analiza, 2nd ed., part 2. Moscow, 1963.
References in periodicals archive ?
It is expected that there will be many applications of the new extension of the classical beta function, e.g., new extension of the beta distribution, new extensions of Gauss hypergeometric functions and confluent hypergeometric function, generating relations, and extension of Riemann-Liouville derivatives.
Slater, Confluent Hypergeometric Functions, Cambridge University Press, 1960.
Geometric properties of special functions such as hypergeometric functions, Bessel functions, Struve functions, Mittag-Leffler functions, Wright functions, and some other related functions are an ongoing part of research in geometric function theory.
whose solutions can be expressed either in terms of confluent hypergeometric functions, as [13]
The corresponding Schrodinger equation is transformed into a Heun equation in canonical form, and exact solutions are obtained in terms of series of hypergeometric functions. The separation of variables is still possible since the vector potential depends only on one spherical variable, [theta] in this case [28].
Mostafa [5] and Porwal and Dixit [6] obtain certain conditions for hypergeometric functions and generalized Bessel functions, respectively, for these classes.
Further relation for the Gauss hypergeometric functions exists as follows:
On the other hand, the fundamental solutions of the second-order equations are much more complex and can be expressed by special functions of mathematical physics, such as the first- and the second-kind Bessel functions of non-integer orders as well as confluent hypergeometric functions [24, 33].
Recently, in [11], the following generalization of (4) has been calculated as a finite sum of terms containing beta and hypergeometric functions, by using the convolution theorem of the Fourier transform.
Historically speaking, a firm footing of the usage of the q-calculus in the context of Geometric Function Theory was actually provided and the basic (or q-) hypergeometric functions were first used in Geometric Function Theory in a book chapter by Srivastava (see, for details, [27]).

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