INFINITE SERIES FORMULAE RELATED TO GAUSS AND BAILEY $_2F_1(\tfrac12)$-SUMS

doi:10.1007/s10473-020-0201-y

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (2): 293-315.doi: 10.1007/s10473-020-0201-y

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INFINITE SERIES FORMULAE RELATED TO GAUSS AND BAILEY $_2F_1(\tfrac12)$-SUMS

初文昌^1,2

1. School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China;
2. Department of Mathematics and Physics, University of Salento(P. O. Box 193), 73100 Lecce, Italy

收稿日期:2017-07-15 修回日期:2019-10-15 出版日期:2020-04-25 发布日期:2020-05-26
作者简介:Wenchang CHU,E-mail:chu.wenchang@unisalento.it

INFINITE SERIES FORMULAE RELATED TO GAUSS AND BAILEY $_2F_1(\tfrac12)$-SUMS

Wenchang CHU^1,2

1. School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China;
2. Department of Mathematics and Physics, University of Salento(P. O. Box 193), 73100 Lecce, Italy

Received:2017-07-15 Revised:2019-10-15 Online:2020-04-25 Published:2020-05-26

摘要/Abstract

摘要： The unified Ω-series of the Gauss and Bailey $_2F_1(\tfrac12)$-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts. Several remarkable transformation theorems for the Ω-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type, including a couple of beautiful expressions for π and the Catalan constant discovered by Guillera (2008).

关键词: Abel's lemma on summation by parts, classical hypergeometric series, Gauss' $_2F_1(\tfrac12)$-sum, Bailey's $_2F_1(\tfrac12)$-sum, Saddle point method, Catalan's constant

Abstract: The unified Ω-series of the Gauss and Bailey $_2F_1(\tfrac12)$-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts. Several remarkable transformation theorems for the Ω-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type, including a couple of beautiful expressions for π and the Catalan constant discovered by Guillera (2008).

Key words: Abel's lemma on summation by parts, classical hypergeometric series, Gauss' $_2F_1(\tfrac12)$-sum, Bailey's $_2F_1(\tfrac12)$-sum, Saddle point method, Catalan's constant

中图分类号:

33C05

初文昌. INFINITE SERIES FORMULAE RELATED TO GAUSS AND BAILEY $_2F_1(\tfrac12)$-SUMS[J]. 数学物理学报(英文版), 2020, 40(2): 293-315.

Wenchang CHU. INFINITE SERIES FORMULAE RELATED TO GAUSS AND BAILEY $_2F_1(\tfrac12)$-SUMS[J]. Acta mathematica scientia,Series B, 2020, 40(2): 293-315.

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INFINITE SERIES FORMULAE RELATED TO GAUSS AND BAILEY $_2F_1(\tfrac12)$-SUMS

INFINITE SERIES FORMULAE RELATED TO GAUSS AND BAILEY $_2F_1(\tfrac12)$-SUMS

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