Fast evaluation of the hypergeometric function pFp-1(a; b; z) at the singular point z = 1 by means of the Hurwitz zeta function ζ(α, s)

Authors:
A. I. Bogolubsky;S. L. Skorokhodov
Affiliations:
Department of Mechanics and Mathematics, Moscow State University, Vorob'evy gory, Moscow, Russia 119992;Dorodnicyn Computing Center, Russian Academy of Sciences, Moscow, Russia 119991
Venue:
Programming and Computing Software
Year:
2006

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Abstract

A method for fast and highly accurate evaluation of the generalized hypergeometric function p F p驴1(a 1, ..., a p ; b 1, ..., b p驴1; 1) = 驴 k = 0 驴 f k by means of the Hurwitz zeta function 驴(驴, s) is developed. Based on asymptotic analysis of the coefficients f k , an expansion of p F p驴1 is constructed as a combination of the functions 驴(驴, s) with explicit coefficients expressed in terms of the generalized Bernoulli polynomials. An appropriate selection of the parameter 驴 and the number of terms of the asymptotic expansion makes it possible to obtain the result with any desired degree of accuracy. The use of computer algebra methods, efficient numerical algorithms, and stochastic optimization methods considerably improve the efficiency of the suggested method.