Functional equations for Hecke-Maass series

The Dirichlet (Hecke-Maass) series associated with the eigenfunctions f and g of the invariant differential operator Delta(k) = -y(2)(partial derivative(2)/partial derivative x(2) + partial derivative(2)/partial derivative y(2)) + iky partial derivative/partial derivative x of weight k are investigated. It is proved that any relation of the form (f \M-k) = g for the k-action of the group SL2(R) is equivalent to a pair of functional equations relating the Hecke-Maass series for f and g and involving only traditional gamma factors.