Fourier transform pairs and Eisenstein-type series related to Jacobi elliptic functions
Peng-Cheng Hang, Alexey Kuznetsov
TL;DR
The paper investigates explicit Fourier transform pairs for ratios of Jacobi elliptic functions, specifically $f(x)=J(2K\'x,k)/\sinh(\pi x)$ or $f(x)=J(2K\'x,k)/\cosh(\pi x)$, and uses Mellin transform techniques to derive sixteen Eisenstein-type double series $\zeta_{j,l}(s,\tau)$. It establishes analytic continuation and functional equations for the completed series $\Lambda_{j,l}(s,\tau)$ under the symmetry $$(s,\tau)\mapsto(1-s,-1/\tau)$$, and provides explicit special values at positive even or odd integers. Fifteen of the sixteen series are obtained from the Fourier transform identities; the remaining case $\Lambda_{0,0}(s,\tau)$ requires a Fourier transform identity (25) connected to the logarithmic derivative of a theta function. The results connect Fourier analysis of elliptic-function transforms with Mellin-analytic structures, elliptic integrals, and a rich set of functional equations, while outlining future work on extending the framework to broader families of Eisenstein-type series and theta-function transforms.
Abstract
We compute Fourier transforms of functions expressed as a ratio of one of the Jacobi elliptic functions divided by $\sinh(πx)$ or $\cosh(πx)$. In many cases, the resulting Fourier transform remains within the same class of functions. Applying the Mellin transform, we obtain sixteen Eisenstein-type series $ζ_{j,l}(s,τ)$, for which we establish several results: analytic continuation with respect to the variable $s$, a functional equation connecting $ζ_{j,l}(s,τ)$ and $ζ_{l,j}(1-s,-1/τ)$, and explicit expressions for $ζ_{j,l}(s,τ)$ when $s$ runs through a sequence of positive even or odd integers.