Hypergeometric series, lemniscate functions, $q$-extensions and Jacobi elliptic functions

Abstract

In this paper, we provide a solution to an open problem posed by Campbell and Chu [12] concerning the explicit evaluation of a constant known as the 'lemniscate-like constant.' We demonstrate that by utilizing tools related to the primary Jacobi elliptic functions, we can derive a closed-form expression in terms of $q-$generalizations of Zeta and Polylogarithm functions and well-known mathematical constants. Lastly, we establish that our primary outcome establishes a non-obvious connection between various and disparate mathematical entities.