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

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.