Barnes multiple zeta-functions, Ramanujan's formula, and relevant series involving hyperbolic functions
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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.
- Journal of the Ramanujan Mathematical Society
Journal of the Ramanujan Mathematical Society 28(1), 49–69-49–69, 2013-03
Ramanujan Mathematical Society