Special Cases
Many of the common mathematical functions can be expressed in terms of the hypergeometric function, or as limiting cases of it. Some typical examples are
The confluent hypergeometric function (or Kummer's function) can be given as a limit of the hypergeometric function
so all functions that are essentially special cases of it, such as Bessel functions, can be expressed as limits of hypergeometric functions. These include most of the commonly used functions of mathematical physics.
Legendre functions are solutions of a second order differential equation with 3 regular singular points so can be expressed in terms of the hypergeometric function in many ways, for example
Several orthogonal polynomials, including Jacobi polynomials P(α,β)
n and their special cases Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials can be written in terms of hypergeometric functions using
Other polynomials that are special cases include Krawtchouk polynomials, Meixner polynomials, Meixner–Pollaczek polynomials.
Elliptic modular functions can sometimes be expressed as the inverse functions of ratios of hypergeometric functions whose arguments a, b, c are 1, 1/2, 1/3, ... or 0. For example, if
then
is an elliptic modular function of τ.
Incomplete beta functions Bx(p,q) are related by
The complete elliptic integrals K and E are given by
Read more about this topic: Hypergeometric Differential Equation
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