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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 (a,b;c;z) are derived for large absolute values of the complex parameters a,b,c (c0,-1,-2,···) and for fixed values of the complex variable z (|arg(1-z)|<π). Assuming a 2 =o(c), b 2 =o(c) and that Re{ a} and Re{ b} are bounded below or two- sided bounded it is shown that the -plane can be divided in two sectors dependent on the value of z so that

(i)F(a,b;c;z) ν=0 (a) ν (b) ν (c) ν z ν ν!,

in the sector including the positive real axis ((a) ν Pochhammer symbol) and F(a,b;c;z)

(ii)πΓ(a+b-c)z 1-c (1-z) c-b-a sin(πc)Γ(1-c)Γ(a)Γ(b) ν=0 (1-a) ν (1-b) ν (1-z) ν (c-b-a+1) ν ν!

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:
33C20Generalized hypergeometric series, p F q
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)