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

2025/10/09 by Hang, Peng-Cheng, Kuznetsov, Alexey
#11M41 #33E05 #42A38 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.08823

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.

Citations

• Asymptotic expansions for the alternating Hurwitz zeta function and its derivatives

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