Reciprocal Hyperbolic Series of Ramanujan Type

Ce Xu and Jianqiang Zhao ()
Additional contact information
Ce Xu: School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China
Jianqiang Zhao: Department of Mathematics, The Bishop’s School, La Jolla, CA 92037, USA

Mathematics, 2024, vol. 12, issue 19, 1-25

Abstract: 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 = F 1 2 ( 1 / 2 , 1 / 2 ; 1 ; x ) and z ? = d z / d x . When a certain parameter in these series is equal to ? , the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.

Keywords: hyperbolic function and trigonometric function; Riemann zeta function; Jacobian elliptic function; Gamma function; residue theorem; Eisenstein series (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/12/19/2974/pdf (application/pdf)
https://www.mdpi.com/2227-7390/12/19/2974/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:12:y:2024:i:19:p:2974-:d:1485324

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().