On the curious series related to the elliptic integrals

TL;DR

This paper introduces a novel summation method for certain series involving hyperbolic functions using elliptic integrals, derivatives, and Mellin transforms, connecting to the Riemann zeta function and Voronoi summation.

Contribution

It develops a new approach to summing series related to elliptic integrals by differentiating with respect to the elliptic modulus and expressing results in closed form.

Findings

Series expressed in terms of complete elliptic integrals

Closed-form solutions for special cases

Connection to Riemann zeta and Voronoi summation

Abstract

By using the theory of the elliptic integrals a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect to the elliptic modulus and integral representations of several of the series in terms of the inverse Mellin transforms related to the Riemann zeta function. The relation with the corresponding case of the Voronoi summation formula is exhibited. The involved series are expressed in closed form in terms of complete elliptic integrals of the first and second kind, and some special cases are calculated in terms of particular values of the Euler gamma function.