Ewald expansions of a class of zeta-functions

The incomplete gamma function expansion for the perturbed Epstein zeta function is known as Ewald expansion. In this paper we state a special case of the main formula in Kanemitsu and Tsukada (Contributions to the theory of zeta-functions: the modular relation supremacy. World Scientific, Singapore, 2014) whose specifications will give Ewald expansions in the H-function hierarchy. An Ewald expansion for us are given by [Formula: see text] or its variants. We shall treat the case of zeta functions which satisfy functional equation with a single gamma factor which includes both the Riemann as well as the Hecke type of functional equations and unify them in Theorem 2. This result reveals the H-function hierarchy: the confluent hypergeometric function series entailing the Ewald expansions. Further we show that some special cases of this theorem entails various well known results, e.g., Bochner–Chandrasekharan theorem, Atkinson–Berndt theorem etc.