  Building on the works of S. Bochner on equivalence of modular relation with
functional equation associated to the Dirichlet series, K. Chandrasekharan and
R. Narasimhan obtained new equivalences between the functional equation and
some arithmetical identities. Sister Ann M. Heath considered the functional
equation in the Hawkins and Knopp context and showed its equivalence to two
arithmetical identities associated with entire modular cusp integrals involving
rational period functions for the full modular group. In this paper we use
techniques of Chandrasekharan and Narasimhan and extend the results of Sister
Ann M. Heath to entire automorphic integrals involving rational period
functions on discrete Hecke group. Moreover, we establish equivalence of two
arithmetical identities with a functional equation associated with automorphic
integrals involving log-polynomial-period functions on the Hecke groups.
