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 Zh. Vychisl. Mat. Mat. Fiz., 2005, Volume 45, Number 4, Pages 574–586 (Mi zvmmf663)

A method for computing the generalized hypergeometric function $_pF_{p-1}(a_1,…,a_p;b_1,…,b_{p-1};1)$ in terms of the riemann zeta function

S. L. Skorokhodov

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

Abstract: A method is proposed for evaluating the generalized hypergeometric function $_pF_{p-1}(a_1,…,a_p;b_1,…,b_{p-1};1)=\sum_{k=0}^\infty f_k$ in terms of the Riemann zeta function $\zeta(s)$ and the Hurwitz zeta function $\zeta(1/2,s)$. By analyzing an asymptotic expansion of the coefficients $f_k$ as $k\to\infty$, an expansion of $_pF_{p-1}$ is constructed in the form of combinations of $\zeta(s)$ and $\zeta(1/2,s)$ with explicit coefficients expressed in terms of generalized Bernoulli polynomials. The convergence of the expansion can be considerably accelerated by choosing optimal values of two control parameters. The efficiency of the method is demonstrated through a great deal of computations and comparisons with Mathematica and Maple.

Key words: generalized hypergeometric function of unit argument, numerical algorithm, Riemann zeta function, Hurwitz zeta function, generalized Bernoulli polynomials.

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English version:
Computational Mathematics and Mathematical Physics, 2005, 45:4, 550–562

Bibliographic databases:
UDC: 519.6:517.588

Citation: S. L. Skorokhodov, “A method for computing the generalized hypergeometric function $_pF_{p-1}(a_1,…,a_p;b_1,…,b_{p-1};1)$ in terms of the riemann zeta function”, Zh. Vychisl. Mat. Mat. Fiz., 45:4 (2005), 574–586; Comput. Math. Math. Phys., 45:4 (2005), 550–562

Citation in format AMSBIB
\Bibitem{Sko05} \by S.~L.~Skorokhodov \paper A method for computing the generalized hypergeometric function ${}_pF_{p-1}(a_1,\dots,a_p;b_1,\dots,b_{p-1};1)$ in terms of the riemann zeta function \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2005 \vol 45 \issue 4 \pages 574--586 \mathnet{http://mi.mathnet.ru/zvmmf663} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2161615} \zmath{https://zbmath.org/?q=an:1077.33008} \elib{http://elibrary.ru/item.asp?id=9139256} \transl \jour Comput. Math. Math. Phys. \yr 2005 \vol 45 \issue 4 \pages 550--562 \elib{http://elibrary.ru/item.asp?id=13489960} 

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This publication is cited in the following articles:
1. Bogolubsky A., Skorokhodov S.L., “Fast evaluation of the hypergeometric function F-p(p-1,)(a; b; z) at the singular point z=1 by means of the Hurwitz zeta function zeta(alpha, s)”, Programming and Computer Software, 32:3 (2006), 145–153
2. Willis J.L., “Acceleration of generalized hypergeometric functions through precise remainder asymptotics”, Numer Algorithms, 59:3 (2012), 447–485
3. Huang Zh.-W., Liu J., “Numexp: Numerical Epsilon Expansion of Hypergeometric Functions”, Comput. Phys. Commun., 184:8 (2013), 1973–1980
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