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 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 Functions
Famous quotes containing the words special and/or cases:
“We cannot set aside an hour for discussion with our children and hope that it will be a time of deep encounter. The special moments of intimacy are more likely to happen while baking a cake together, or playing hide and seek, or just sitting in the waiting room of the orthodontist.”
—Neil Kurshan (20th century)
“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 (18641929)