On a generating function of Niebur-Poincar\'e series

Let $\Gamma\subset PSL_2(\mathbb{R})$ 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--Poincar\'e series $\{F_{m,s}(\tau)\}_{m\ge 1}$ 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 Poincar\'e type series involving polylogarithms. We also consider a generating function formed with derivatives in $s$ of the Niebur--Poincar\'e series and prove that the continuation of the generating function at $s=1$ can be expressed in terms of $\Gamma$-periodization of a point-pair invariant involving the Rogers dilogarithm and the Kronecker limit function associated to the non-holomorphic Eisenstein series.