General Hypergeometric function can be expressed in power series

where is Pochhammar symbol,

The General hypergeometric function satisfies the following differential equation,

where

For , the differential equation becomes

For ,

For

For

The Gauss Hypergeometric function is ,

which satisfies,

There are some interesting expression for Pochhammar symbol

when

when

Here are list of common function into hypergeometric function

where is Bessel function of first kind, which satisfies

where is modified Bessel function of first kind, which satisfies

where is error function

where is Legendre function, which satisfies

where is associate Legendre function, which satisfies

where is complete elliptic integral of 1st kind

and is complete elliptic integral of 2nd kind

Reference

“Notes on hypergeometric functions” by John D. Cook (April 10, 2003)

“Generalized Hypergeometric Series” by W. N. Bailey, Cambridge (1935)

“Handbook of Mathematical Functions” by Abramowitz and Stegun (1964)

“The special functions and their approximations” by Yudell L. Luke v. 1 (1969)

“Concrete Mathematics” by Graham, Knuth, and Patashnik (1994)

In Wolfram research (http://functions.wolfram.com/functions.html), many functions are listed. We can click to a function, then we click “Representations through more general functions”, then “Through hypergeometric functions”, then we can see how the function looks like.