2025/12/15 by Kathrin Bringmann, Bringmann, Kathrin, Jay Jorgenson +3
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities
paper · pdf · doi:10.48550/arxiv.2512.13167
Let Γ⊂ PSL2(ℝ) be a Fuchsian group of the first kind which has a cusp i∞ of width one. In this paper, we first consider a generating function formed with the Niebur--Poincaré series \Fm,s(τ)\m≥ 1 associated to i∞. 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∞, also at s=1. Secondly, we show that, for any s∈ 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.