Generalized Addition Theorems for the Jacobi Elliptic Functions

Abstract

In this paper, we generalize the classical addition theorems for the Jacobi elliptic functions, giving numerous implicit and explicit formulas. We first present determinant identities involving several of the Jacobi functions. Then, we prove explicit addition formulas for a single function. Novel 4-,5-,6- and 7-term addition formulas are also given alongside a general $n$-term addition formula.