  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.
