1. Introduction. Implicit in the
work of Rankin [6] and explicit in the work of Petersson [5] is a formula connecting
the Petersson inner product of two holomorphic modular forms with a residue of the
Dirichlet series formed with the products of the Fourier coefficients of the two
modular forms. In view of the modern group theoretic interpretation of the
eigenfunctions of the Hecke operators as unitary representations of an adèle group,
it appears that the ideas of Rankin and Petersson may have wider applicability; for
example they may relate to multiplicity-one problems in the theory of automorphic
representations. The purpose of this note is to extend these ideas to real analytic
modular forms.
