Reciprocal Hyperbolic Series of Ramanujan Type

被引：0

作者：

Xu, Ce ^{[1
]}

Zhao, Jianqiang ^{[2
]}

机构：

[1] Anhui Normal Univ, Sch Math & Stat, Wuhu 241002, Peoples R China

[2] Bishops Sch, Dept Math, La Jolla, CA 92037 USA

来源：

MATHEMATICS | 2024年 / 12卷 / 19期

基金：

中国国家自然科学基金;

关键词：

hyperbolic function and trigonometric function; Riemann zeta function; Jacobian elliptic function; Gamma function; residue theorem; Eisenstein series; EISENSTEIN SERIES; INFINITE SERIES; SUMMATION; FORMULAS; ANALOGS;

D O I：

10.3390/math12192974

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z=F12(1/2,1/2;1;x) and z '=dz/dx. When a certain parameter in these series is equal to pi, the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.

引用

页数：25

共 40 条

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[3]

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[4]

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[5]

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