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Zbl 0732.33005
Wagner, E.
Asymptotische Entwicklungen der Gaussschen hypergeometrischen Funktion für unbeschränkte Parameter. (Asymptotic expansions of the Gauss hypergeometric function for unbounded parameters).
(German)
[J] Z. Anal. Anwend. 9, No.4, 351-360 (1990). ISSN 0232-2064; ISSN 1661-4534/e

Asymptotic expansions of the Gauss hypergeometric function ${}\sb 2F\sb 1(a,b;c;z)$ are derived for large absolute values of the complex parameters a,b,c (c$\ne 0,-1,-2,...)$ and for fixed values of the complex variable z $(\vert \arg (1-z)\vert <\pi)$. Assuming $a\sp 2=o(c)$, $b\sp 2=o(c)$ and that Re$\{$ $a\}$ and Re$\{$ $b\}$ are bounded below or two- sided bounded it is shown that the ${\bbfC}$-plane can be divided in two sectors dependent on the value of z so that $$(i)\quad F(a,b;c;z)\approx \sum\sp{\infty}\sb{\nu =0}\frac{(a)\sb{\nu}(b)\sb{\nu}}{(c)\sb{\nu}}\frac{z\sp{\nu}}{\nu !},$$ in the sector including the positive real axis $((a)\sb{\nu}$ Pochhammer symbol) and F(a,b;c;z)$\approx$ \par $$(ii)\quad \approx \frac{\pi \Gamma (a+b-c)z\sp{1-c}(1-z)\sp{c-b- a}}{\sin (\pi c)\Gamma (1-c)\Gamma (a)\Gamma (b)}\sum\sp{\infty}\sb{\nu =0}\frac{(1-a)\sb{\nu}(1-b)\sb{\nu}(1-z)\sp{\nu}}{(c-b-a+1)\sb{\nu}\nu !}$$ in the remaining sector. In particular, it follows that (i) is not valid for all z with $\vert z\vert <1$, when the complex parameter c tends arbitrary to infinity. This refutes an assertion in the well-known book Higher transcendental functions'', Vol. 1 by {\it A. Erdélyi} et. al. (1951; Zbl 0051.303).
[E.Wagner]
MSC 2000:
*33C20 Generalized hypergeometric series
41A60 Asymptotic problems in approximation

Citations: Zbl 0051.303

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