Journal of the Ramanujan Mathematical Society

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Barnes Multiple Zeta-Functions, Ramanujan’s Formula, and Relevant Series Involving Hyperbolic Functions


Affiliations
1 Department of Mathematics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan
2 Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
3 3Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan
     

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In the former part of this paper, we give functional equations for Barnes multiple zeta-functions and consider some relevant results. In particular, we show that Ramanujan’s classical formula for the Riemann zeta values can be derived from functional equations for Barnes zetafunctions. In the latter half part, we generalize some evaluation formulas for certain series involving hyperbolic functions in terms of Bernoulli polynomials. The original formulas were classically given by Cauchy, Mellin, Ramanujan, and later recovered and reformulated by Berndt. From our consideration, we give multiple versions of these known formulas.
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  • Barnes Multiple Zeta-Functions, Ramanujan’s Formula, and Relevant Series Involving Hyperbolic Functions

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Authors

Yasushi Komori
Department of Mathematics, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan
Kohji Matsumoto
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
Hirofumi Tsumura
3Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan

Abstract


In the former part of this paper, we give functional equations for Barnes multiple zeta-functions and consider some relevant results. In particular, we show that Ramanujan’s classical formula for the Riemann zeta values can be derived from functional equations for Barnes zetafunctions. In the latter half part, we generalize some evaluation formulas for certain series involving hyperbolic functions in terms of Bernoulli polynomials. The original formulas were classically given by Cauchy, Mellin, Ramanujan, and later recovered and reformulated by Berndt. From our consideration, we give multiple versions of these known formulas.