Kronecker limit functions and an extension of the Rohrlich-Jensen formula

In 1984 Rohrlich proved a modular analogue of Jensen's formula. Under certain conditions, the Rohrlich-Jensen formula expresses an integral of the log-norm $\log \Vert f \Vert$ of a $\text{\rm PSL}(2,\ZZ)$ modular form $f$ in terms of the Dedekind Delta function evaluated at the divisor of $f$. Recently, Bringmann-Kane re-interpreted the Rohrlich-Jensen formula as evaluating a regularized inner product of $\log \Vert f \Vert$ and extended the result to compute a regularized inner product of $\log \Vert f \Vert$ with what amounts to powers of the Hauptmoduli of $\text{\rm PSL}(2,\ZZ)$. In the present article, we revisit the Rohrlich-Jensen formula and prove that it can be viewed as a regularized inner product of special values of two Poincar\'e series, one of which is the Niebur-Poincar\'e series and the other is the resolvent kernel of the Laplacian. The regularized inner product can be seen as a type of Maass-Selberg relation. In this form, we develop a Rohrlich-Jensen formula associated to any Fuchsian group $\Gamma$ of the first kind with one cusp by employing a type of Kronecker limit formula associated to the resolvent kernel. We present two examples of our main result: First, when $\Gamma$ is the full modular group $\text{\rm PSL}(2,\ZZ)$, thus reproving the theorems from \cite{BK19}; and second when $\Gamma$ is an Atkin-Lehner group $\Gamma_{0}(N)^+$, where explicit computations are given for certain genus zero, one and two levels.