Sums Involving the Hurwitz Zeta Function |
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Authors: | Kanemitsu Shigeru Kumagai Hiroshi Yoshimoto Masami |
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Institution: | (1) Department of Electrical Engineering, University of Kinki, Iizuka, Fukuoka, 820-0011, Japan;(2) Kagoshima National College of Technology, 1460-1 Shinko Hayato-cho, 899-5193, Japan;(3) Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan |
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Abstract: | We shall prove a general closed formula for integrals considered by Ramanujan, from which we derive our former results on sums involving Hurwitz zeta-function in terms not only of the derivatives of the Hurwitz zeta-function, but also of the multiple gamma function, thus covering all possible formulas in this direction. The transition from the derivatives of the Hurwitz zeta-function to the multiple gamma function and vice versa is proved to be effected essentially by the orthogonality relation of Stirling numbers. |
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Keywords: | Hurwitz zeta-function multiple gamma function Stirling numbers |
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