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Reciprocal Hyperbolic Series of Ramanujan Type

Author

Listed:
  • Ce Xu

    (School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, China
    These authors contributed equally to this work.)

  • Jianqiang Zhao

    (Department of Mathematics, The Bishop’s School, La Jolla, CA 92037, USA
    These authors contributed equally to this work.)

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 π , the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.

Suggested Citation

  • Ce Xu & Jianqiang Zhao, 2024. "Reciprocal Hyperbolic Series of Ramanujan Type," Mathematics, MDPI, vol. 12(19), pages 1-25, September.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:19:p:2974-:d:1485324
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