  Marvin Knopp developed the theory of automorphic integrals, which generalize
automorphic forms; each automorphic integral has an additional period function
in its automorphic relation. The period functions satisfy relations that arise
from the underlying group relations. Knopp showed that entire automorphic
integrals with rational period functions satisfy a Hecke correspondence
theorem, provided the rational period functions have poles only at 0 or
infinity. For other automorphic integrals each corresponding Dirichlet series
has a functional equation with a remainder term that arises from the nonzero
poles of the rational period function.
  In this paper we prove a Hecke correspondence theorem for a class of
automorphic integrals with rational period functions on the Hecke groups. We
restrict our attention to automorphic integrals of weight that is twice an odd
integer and to rational period functions that satisfy a symmetry property we
call "Hecke-symmetry." We explicate the relationship between the structure of
the rational period functions and the corresponding remainder terms. Each
remainder term satisfies two relations (the second of which is new in this
paper) corresponding to the two relations for the rational period function.
