| Summary: | 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 \documentclass[12pt]{minimal}
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\begin{document}$$H_{1,2}^{2,0}\leftrightarrow H_{1,2}^{1,1}$$\end{document}H1,22,0↔H1,21,1 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.
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