Two-dimensional series evaluations via the elliptic functions of Ramanujan and Jacobi

TL;DR

This paper presents the first closed-form evaluations of specific double series resembling lattice sums, utilizing elliptic functions, singular moduli, class invariants, and the Rogers-Ramanujan continued fraction.

Contribution

It introduces novel closed-form formulas for double series using advanced elliptic function theory and Ramanujan-Jacobi functions.

Findings

First closed-form evaluations of certain double series

Connections established between lattice sums and elliptic functions

New identities involving Rogers-Ramanujan continued fraction

Abstract

We evaluate in closed form, for the first time, certain classes of double series, which are remindful of lattice sums. Elliptic functions, singular moduli, class invariants, and the Rogers--Ramanujan continued fraction play central roles in our evaluations