Hypergeometric Differential Equation - Special Cases

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

\begin{align} \ln(1+z) &= z\, _2F_1(1,1;2;-z) \\ (1-z)^{-a} &= \, _2F_1(a,1;1;z) \\ \arcsin(z) &= z \, _2F_1\left(\tfrac{1}{2}, \tfrac{1}{2}; \tfrac{3}{2};z^2\right). \end{align}

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

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