title: Ewald expansions of a class of zeta-functions.
abstract: 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.
authors.affiliation: Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad, UP 211019 India.
authors.forename: Kalyan
authors.initials: K
authors.lastname: Chakraborty
authors.affiliation: Graduate School of Advanced Technology, Kinki University, Iizuka, Fukuoka 820-8555 Japan.
authors.forename: Shigeru
authors.identifier: 0000-0002-8489-7665
authors.initials: S
authors.lastname: Kanemitsu
authors.affiliation: Graduate School of Advanced Technology, Kinki University, Iizuka, Fukuoka 820-8555 Japan.
authors.forename: Haruo
authors.initials: H
authors.lastname: Tsukada
keywords: Ewald expansion; Functional equation; Modular relation; Zeta-function