Series expansions for Maass forms on the full modular group from the Farey transfer operators

TL;DR

This paper develops new series expansions for Maass forms on the modular surface using transfer operators related to Farey maps, enhancing understanding of their Fourier coefficients and functional equations.

Contribution

It introduces an inverse integral transform to derive novel series expansions for Maass cusp forms and Eisenstein series, connecting transfer operators with Maass form analysis.

Findings

New series expansions for Maass cusp forms

Enhanced understanding of Fourier coefficients

A novel series expansion for the divisor function

Abstract

We deepen the study of the relations previously established by Mayer, Lewis and Zagier, and the authors, among the eigenfunctions of the transfer operators of the Gauss and the Farey maps, the solutions of the Lewis-Zagier three-term functional equation and the Maass forms on the modular surface $PSL(2,\Z)\backslash \HH$. In particular we introduce an "inverse" of the integral transform studied by Lewis and Zagier, and use it to obtain new series expansions for the Maass cusp forms and the non-holomorphic Eisenstein series restricted to the imaginary axis. As corollaries we obtain further information on the Fourier coefficients of the forms, including a new series expansion for the divisor function.