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))