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

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)H. 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, and prove the analogous of the Lindelo ̈f hypothesis for the behaviour of the non-holomorphic Eisenstein series on the imaginary axis.