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 ?
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.
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2](x, y) can be expressed in terms of hypergeometric functions of x/[(1 - 4y).
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Although the exact solution is expressed in the form of standard and generalized hypergeometric functions, it can be easily incorporated in modern computational environments like Mathematica.
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Some topics covered include two-dimensional directed lattice walks with boundaries, partition polynomials, hypergeometric functions related to series acceleration formulas, and using integer relation algorithms for finding relationships among functions.
Hypergeometric functions arise when integrating this, but we avoid these by expanding ([R-[R.
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