Fourier transform pairs and Eisenstein-type series related to Jacobi elliptic functions
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910037446754304 |
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| author | Hang, Peng-Cheng Kuznetsov, Alexey |
| author_facet | Hang, Peng-Cheng Kuznetsov, Alexey |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08823 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier transform pairs and Eisenstein-type series related to Jacobi elliptic functions Hang, Peng-Cheng Kuznetsov, Alexey Classical Analysis and ODEs Number Theory 33E05, 42A38, 11M41 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. |
| title | Fourier transform pairs and Eisenstein-type series related to Jacobi elliptic functions |
| topic | Classical Analysis and ODEs Number Theory 33E05, 42A38, 11M41 |
| url | https://arxiv.org/abs/2510.08823 |