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

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

2010-06-16

arxiv.org/abs/1006.3336v1

Mathematics

Number Theory

16 pages

Scientific paper

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 of certain series involving hyperbolic functions in terms of Bernoulli polynomials. The original formulas were classically given by Cauchy, Mellin, Ramanujan, and later recovered and formulated by Berndt. From our consideration, we give multiple versions of these known formulas.

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