Journal of the Ramanujan Mathematical Society

Volume 28, Issue 1, March 2013 pp. 49--69.

Barnes multiple zeta-functions, Ramanujan's formula, and relevant series involving hyperbolic functions

Authors: Yasushi Komori, Kohji Matsumoto and Hirofumi Tsumura
Author institution: Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1, Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan

Summary: 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 zeta-functions. 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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