Asymptotische Darstellungen der hypergeometrischen Funktionen für grosse Werte eines Parameters

Abstract

Asymptotic expansions of the hypergeometric function $F(a, b, c; z)$ for $|b| \to \infty$ where $a, c, z$ are fixed complex numbers given in well-known tables (e.g. Bateman/Erdélyi: Higher transcendental functions I. New York 1953) are incorrect. In the present paper asymptotic representations of the hypergeomctric function $F$ for $a$ (complex) $\to \infty$ are derived where $b, c (c \neq 0, -1, -2,\dots)$ and $z (z \neq 0$, |Arg $(1- z) | < \pi$ are fixed complex numbers. By change of $a$ and $b$ appropriate asymptotic representations of $F$ for $|b| \to \infty$ are obtained.