On a generating function of Niebur-Poincaré series
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909962992615424 |
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| author | Bringmann, Kathrin Jorgenson, Jay Smajlović, Lejla |
| author_facet | Bringmann, Kathrin Jorgenson, Jay Smajlović, Lejla |
| contents | Let $Γ\subset PSL_2(\mathbb{R})$ be a Fuchsian group of the first kind which has a cusp $i\infty$ of width one. In this paper, we first consider a generating function formed with the Niebur--Poincaré series $\{F_{m,s}(τ)\}_{m\ge 1}$ associated to $i\infty$. We prove a relation between the continuation of this generating function to $s=1$ with the resolvent kernel associated to the hyperbolic Laplacian and the non-holomorphic Eisenstein series associated to $i\infty$, also at $s=1$. Secondly, we show that, for any $s\in \mathbb{N}$, the generating function equals Poincaré type series involving polylogarithms. We also consider a generating function formed with derivatives in $s$ of the Niebur--Poincaré series and prove that the continuation of the generating function at $s=1$ can be expressed in terms of $Γ$-periodization of a point-pair invariant involving the Rogers dilogarithm and the Kronecker limit function associated to the non-holomorphic Eisenstein series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13167 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a generating function of Niebur-Poincaré series Bringmann, Kathrin Jorgenson, Jay Smajlović, Lejla Number Theory 11F12, 11F37 Let $Γ\subset PSL_2(\mathbb{R})$ be a Fuchsian group of the first kind which has a cusp $i\infty$ of width one. In this paper, we first consider a generating function formed with the Niebur--Poincaré series $\{F_{m,s}(τ)\}_{m\ge 1}$ associated to $i\infty$. We prove a relation between the continuation of this generating function to $s=1$ with the resolvent kernel associated to the hyperbolic Laplacian and the non-holomorphic Eisenstein series associated to $i\infty$, also at $s=1$. Secondly, we show that, for any $s\in \mathbb{N}$, the generating function equals Poincaré type series involving polylogarithms. We also consider a generating function formed with derivatives in $s$ of the Niebur--Poincaré series and prove that the continuation of the generating function at $s=1$ can be expressed in terms of $Γ$-periodization of a point-pair invariant involving the Rogers dilogarithm and the Kronecker limit function associated to the non-holomorphic Eisenstein series. |
| title | On a generating function of Niebur-Poincaré series |
| topic | Number Theory 11F12, 11F37 |
| url | https://arxiv.org/abs/2512.13167 |