Volume 25, pp. 115-120, 2006.

An integral representation of some hypergeometric functions

K. A. Driver and S. J. Johnston

Abstract

The Euler integral representation of the ${}_2{F}_1$ Gauss hypergeometric function is well known and plays a prominent role in the derivation of transformation identities and in the evaluation of ${}_2{F}_1(a,b;c;1)$, among other applications. The general ${}_{p+k}{F}_{q+k}$ hypergeometric function has an integral representation where the integrand involves ${}_p{F}_q$. We give a simple and direct proof of an Euler integral representation for a special class of ${}_{q+1}{F}_q$ functions for $q\ge 2$. The values of certain ${}_3{F}_2$ and ${}_4{F}_3$ functions at $x=1$, some of which can be derived using other methods, are deduced from our integral formula.

Full Text (PDF) [121 KB], BibTeX

Key words

3F2 hypergeometric functions, general hypergeometric functions, integral representation

AMS subject classifications

15A15