Fourier transform pairs and Eisenstein-type series related to Jacobi elliptic functions

Abstract

We compute Fourier transforms of functions expressed as a ratio of one of the Jacobi elliptic functions divided by $\sinh(\pi x)$ or $\cosh(\pi x)$. In many cases, the resulting Fourier transform remains within the same class of functions. Applying the Mellin transform, we obtain sixteen Eisenstein-type series $\zeta_{j,l}(s,\tau)$, for which we establish several results: analytic continuation with respect to the variable $s$, a functional equation connecting $\zeta_{j,l}(s,\tau)$ and $\zeta_{l,j}(1-s,-1/\tau)$, and explicit expressions for $\zeta_{j,l}(s,\tau)$ when $s$ runs through a sequence of positive even or odd integers.