Hypergeometric Function - 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

.
(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)

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 examples, 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 Function

Famous quotes containing the words special and/or cases:

    “... there has been a very special man in my life for the past year. All I’ll say about him is that he’s kind, warm, mature, someone I can trust—and he’s not a politician.”
    —Donna Rice (b. c. 1962)

    “In most cases a favorite writer is more with us in his book than he ever could have been in the flesh; since, being a writer, he is one who has studied and perfected this particular mode of personal incarnation, very likely to the detriment of any other. I should like as a matter of curiosity to see and hear for a moment the men whose works I admire; but I should hardly expect to find further intercourse particularly profitable.”
    —Charles Horton Cooley (1864–1929)