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Reciprocal Hyperbolic Series of Ramanujan Type

Ce Xu, Jianqiang Zhao

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 $z={}_2F_1(1/2,1/2;1;x)$ and $z'=dz/dx$. 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.

Reciprocal Hyperbolic Series of Ramanujan Type

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 and . 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.

Paper Structure

This paper contains 13 sections, 14 theorems, 100 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Some Lemmas and Transformations
  3. Three Lemmas
  4. Three Transformations
  5. Evaluations of some reciprocal hyperbolic series
  6. Some preliminary results
  7. Evaluations of ${\bar{S}}_{2p,2}(y),\ {\bar{S}}'_{2p-1,2}(y),\ {\bar{C}}_{2p,2}(y)\ {\rm and}\ {\bar{C}}'_{2p-1,2}(y)$
  8. Evaluations of $S_{2p,2}(y),\ { S}'_{2p,2}(y),\ C_{2p,2}(y)\ {\rm and}\ { C}'_{2p,2}(y)$
  9. Three Differential Equations
  10. Examples and Further Results
  11. Examples
  12. Further Results
  13. Concluding Remarks

Key Result

Lemma 2.1

(Residue Theorem, FS1998) Let $\xi (s)$ be a kernel function and let $r(s)$ be a function which is $O(s^{-2})$ at infinity. Then where $S$ is the set of poles of $r(s)$ and $T$ is the set of poles of $\xi(s)$ that are not poles $r(s)$. Here $\underset{s=\alpha}\mathop{\mathrm{Res}}\nolimits (r(s))$ denotes the residue of $r(s)$ at $s=\alpha$. The kernel function $\xi(s)$ is meromorphic in the who

Theorems & Definitions (31)

  • Definition 1.1
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • proof
  • Definition 3.1
  • Theorem 3.1
  • ...and 21 more