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 $b,c(c\ne 0,-1,-2,\dots )$ and $z(z\ne 0$, |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.