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We introduce a method that is based on Fourier series expansions related toJacobi elliptic functions and that we apply to determine new identities forevaluating 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 producingevaluations for hyperbolic sums with sign functions that have not previouslyappeared in the literature on hyperbolic sums. We apply our method using theJacobian elliptic functions $\text{dc}$ and $\text{nc}$, together with theelliptic alpha function, to obtain new closed forms for $q$-digammaexpressions, and new closed forms for series related to discoveries due toRamanujan, Berndt, and others.

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