Hyperbolic summations derived using the Jacobi functions $\text{dc}$ and $\text{nc}$

TL;DR

This paper introduces a novel method using Fourier series and Jacobi elliptic functions to derive new identities for hyperbolic sums, generalizing classical results and providing new closed-form evaluations involving elliptic integrals and special functions.

Contribution

The paper develops a new approach based on Fourier series and Jacobi functions to evaluate hyperbolic sums, extending Ramanujan and Zucker's work with novel sign-function sums and closed-form formulas.

Findings

Derived new identities for hyperbolic sums in terms of elliptic integrals.

Generalized Ramanujan and Zucker's hyperbolic sum formulas.

Obtained new closed-form expressions for q-digamma and related series.

Abstract

We introduce a method that is based on Fourier series expansions related to Jacobi elliptic functions and that we apply to determine new identities for evaluating hyperbolic infinite sums in terms of the complete elliptic integrals $K$ and $E$. We apply our method to determine generalizations of a family of $\text{sech}$-sums given by Ramanujan and generalizations of a family of $\text{csch}$-sums given by Zucker. Our method has the advantage of producing evaluations for hyperbolic sums with sign functions that have not previously appeared in the literature on hyperbolic sums. We apply our method using the Jacobian elliptic functions $\text{dc}$ and $\text{nc}$, together with the elliptic alpha function, to obtain new closed forms for $q$-digamma expressions, and new closed forms for series related to discoveries due to Ramanujan, Berndt, and others.