# Hypergeometric Functions

## 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 ?
Gupta, "Extended matrix variate hypergeometric functions and matrix variate distributions," Int.
Shaffer, Starlike and prestarlike hypergeometric functions, SIAM Journal on Mathematical Analysis, vol.
Kita, Theory of hypergeometric functions, translated from the Japanese by Kenji Iohara, Springer Monographs in Mathematics, Springer, Tokyo, 2011.
Acharya, Univalence criteria for analytic functions and applications to hypergeometric functions, Ph.
1]; as the remaining integral in that equation could be evaluated analytically into a complicated expression involving hypergeometric functions, we preferred to calculate the integral numerically.
He begins by discussing generalized hypergeometric functions, of which Bessel functions are a special case, then applies the results to many useful representation of Bessel functions and their integrals by specializing the parameters.
Saigo, "A remark on integral operators involving the Gauss hypergeometric functions," Mathematical Reports, College of General Education, Kyushu University, vol.
Owa, Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, hadamard products, linear operators, and certain subclasses of analytic functions, Nagoya Math.
Srivastava, Univalent and starlike generalised hypergeometric functions, Canad.
2](x, y) can be expressed in terms of hypergeometric functions of x/[(1 - 4y).
and these functions can be represented in terms of the hypergeometric functions and their explicit expressions are given in the paper [34].

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