On the generalized Dirichlet beta and Riemann zeta functions and Ramanujan-type formulae for beta and zeta values

Abstract

We define the generalized Dirichlet beta and Riemann zeta functions in terms of the integrals, involving powers of the hyperbolic secant and cosecant functions. The corresponding functional equations are established. Some consequences of the Ramanujan identity for zeta values at odd integers are investigated and new formulae of the Ramanujan type are obtained. These results are achieved, in particular, involving the Kontorovich-Lebedev transform and the corresponding polynomials introduced by the author.