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Asymptotische Entwicklungen der Gaußschen hypergeometrischen Funktion für unbeschränkte Parameter. (Asymptotic expansions of the Gauss hypergeometric function for unbounded parameters). (German) Zbl 0732.33005

Asymptotic expansions of the Gauss hypergeometric function ${}_{2}{F}_{1}\left(a,b;c;z\right)$ are derived for large absolute values of the complex parameters a,b,c (c$\ne 0,-1,-2,···\right)$ and for fixed values of the complex variable z $\left(|arg\left(1-z\right)|<\pi \right)$. Assuming ${a}^{2}=o\left(c\right)$, ${b}^{2}=o\left(c\right)$ and that Re$\left\{$ $a\right\}$ and Re$\left\{$ $b\right\}$ are bounded below or two- sided bounded it is shown that the $ℂ$-plane can be divided in two sectors dependent on the value of z so that

$\left(i\right)\phantom{\rule{1.em}{0ex}}F\left(a,b;c;z\right)\approx \sum _{\nu =0}^{\infty }\frac{{\left(a\right)}_{\nu }{\left(b\right)}_{\nu }}{{\left(c\right)}_{\nu }}\frac{{z}^{\nu }}{\nu !},$

in the sector including the positive real axis $\left({\left(a\right)}_{\nu }$ Pochhammer symbol) and F(a,b;c;z)$\approx$

$\left(ii\right)\phantom{\rule{1.em}{0ex}}\approx \frac{\pi {\Gamma }\left(a+b-c\right){z}^{1-c}{\left(1-z\right)}^{c-b-a}}{sin\left(\pi c\right){\Gamma }\left(1-c\right){\Gamma }\left(a\right){\Gamma }\left(b\right)}\sum _{\nu =0}^{\infty }\frac{{\left(1-a\right)}_{\nu }{\left(1-b\right)}_{\nu }{\left(1-z\right)}^{\nu }}{{\left(c-b-a+1\right)}_{\nu }\nu !}$

in the remaining sector. In particular, it follows that (i) is not valid for all z with $|z|<1$, when the complex parameter c tends arbitrary to infinity. This refutes an assertion in the well-known book “Higher transcendental functions”, Vol. 1 by A. Erdélyi et. al. (1951; Zbl 0051.303).

Reviewer: E.Wagner
##### MSC:
 33C20 Generalized hypergeometric series, ${}_{p}{F}_{q}$ 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)