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

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Abstract

A method for fast and highly accurate evaluation of the generalized hypergeometric function p F p−1(a 1, ..., a p ; b 1, ..., b p−1; 1) = Σ ∞ k = 0 f k by means of the Hurwitz zeta function ζ(α, s) is developed. Based on asymptotic analysis of the coefficients f k , an expansion of p F p−1 is constructed as a combination of the functions ζ(α, s) with explicit coefficients expressed in terms of the generalized Bernoulli polynomials. An appropriate selection of the parameter α 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.

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, Vorob’evy gory, Moscow, 119992, Russia

    A. I. Bogolubsky

  2. Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119991, Russia

    S. L. Skorokhodov

Authors
  1. A. I. Bogolubsky
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  2. S. L. Skorokhodov
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Original Russian Text © A.I. Bogolubsky, S.L. Skorokhodov, 2006, published in Programmirovanie, 2006, Vol. 32, No. 3.

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Bogolubsky, A.I., Skorokhodov, S.L. Fast evaluation of the hypergeometric function p F p−1(a; b; z) at the singular point z = 1 by means of the Hurwitz zeta function ζ(α, s). Program Comput Soft 32, 145–153 (2006). https://doi.org/10.1134/S0361768806030054

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  • DOI: https://doi.org/10.1134/S0361768806030054

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