Volume 25, pp. 115-120, 2006.

An integral representation of some hypergeometric functions

K. A. Driver and S. J. Johnston


The Euler integral representation of the ${}_2F_1$ Gauss hypergeometric function is well known and plays a prominent role in the derivation of transformation identities and in the evaluation of ${}_2F_1(a,b;c;1)$, among other applications. The general ${}_{p+k}F_{q+k}$ hypergeometric function has an integral representation where the integrand involves ${}_pF_q$. We give a simple and direct proof of an Euler integral representation for a special class of ${}_{q+1}F_q$ functions for $q\geq 2$. The values of certain ${}_3F_2$ and ${}_4F_3$ functions at $x=1$, some of which can be derived using other methods, are deduced from our integral formula.

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Key words

3F2 hypergeometric functions, general hypergeometric functions, integral representation

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