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Special value of the hypergeometric function ${}_{3}{F}_{2}$ and connection formulae among asymptotic expansions. (English) Zbl 0672.33002
Kummer was the first to realize that ${}_{3}{F}_{2}\left(a,b,c;d,e;x\right)$ is not nearly as nice as ${}_{2}{F}_{1}\left(a,b;d;x\right)$, but that the ${}_{3}{F}_{2}$ with $x=1$ is very close to the ${}_{2}{F}_{1}$ at x. The present paper is a very important study of the general ${}_{3}{F}_{2}$ at $x=1$. Among other facts there is an analytic continuation when the series does not converge, and the asymptotic behavior as some or all the parameters grow linearly. One of the essential tools is an integral representation over a simplex and its regularization.
##### MSC:
 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$
##### Keywords:
asymptotic behavior; connection coefficients