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 It might be outdated or ideologically biased.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))


Whittaker, E. T., and G. N. Watson. Kurs sovremennogo analiza, 2nd ed., part 2. Moscow, 1963.
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
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Slater, Confluent Hypergeometric Functions, Cambridge University Press, 1960.
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