On q-series and the moment problem associated to local factors

We investigate the moment problem and Jacobi matrix associated -- by the operator theoretic framework of the semilocal trace formula -- to each finite set $S$ of places of $\mathbb Q$ containing the archimedean place. The measure is given by the absolute value squared of the product over $S$ of local factors restricted to the critical line. We treat the case $S=\{p,\infty\}$, where a single prime $p$ is adjoined to the archimedean place. We find that all the key ingredients such as the moments, the orthogonal polynomials and the Jacobi matrices can be expressed as power series in terms of the parameter $q:=1/p$. We show that the series which appear for the moments themselves are Lambert series. The study of the $q$-series for the coefficients of the Jacobi matrix, and for the associated orthogonal polynomials reveals an intriguing integrality result: all those coefficients belong to the ring $\mathbb Z[\frac{1}{\sqrt{2}}]$ obtained by adjoining $1 / \sqrt{2}$ to the ring of integers. The main result of this paper is the conceptual explanation of this integrality property using Catalan numbers.