Asymptotics of a Gauss hypergeometric function with large parameters, IV: A uniform expansion

2018/09/24 by R B Paris, Paris, R B
Mathematics · #33C05 #34E05 #41A60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:33C05 #msc:34E05 #msc:41A60

paper · pdf · doi:10.48550/arxiv.1809.08794

9 pages, 0 figures

arxiv created 2018/10/14 · arxiv updated 2018/10/16

Abstract

We consider the uniform asymptotic expansion for the Gauss hypergeometric function F(a+ελ,m;c+λ;x), λ→+∞ for x<1 and positive integer m when the parameter ε>1 and the constants a and c are supposed finite. When m=1, we employ the standard procedure of the method of steepest descents modified to deal with the situation when a saddle point is near a simple pole. It is shown that it is possible to give a closed-form expression for the coefficients in the resulting uniform expansion. The expansion when m≥ 2 is obtained by means of a recurrence relation. Numerical results illustrating the accuracy of the resulting expansion are given.

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