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Marvin Knopp developed the theory of automorphic integrals, which generalizeautomorphic forms; each automorphic integral has an additional period functionin its automorphic relation. The period functions satisfy relations that arisefrom the underlying group relations. Knopp showed that entire automorphicintegrals with rational period functions satisfy a Hecke correspondencetheorem, provided the rational period functions have poles only at 0 orinfinity. For other automorphic integrals each corresponding Dirichlet serieshas a functional equation with a remainder term that arises from the nonzeropoles of the rational period function.  In this paper we prove a Hecke correspondence theorem for a class ofautomorphic integrals with rational period functions on the Hecke groups. Werestrict our attention to automorphic integrals of weight that is twice an oddinteger and to rational period functions that satisfy a symmetry property wecall "Hecke-symmetry." We explicate the relationship between the structure ofthe rational period functions and the corresponding remainder terms. Eachremainder term satisfies two relations (the second of which is new in thispaper) corresponding to the two relations for the rational period function.

A Hecke Correspondence Theorem for Automorphic Integrals with Symmetric Rational Period Functions on the Hecke Groups