Ramaswami Type translation formulae for the polylogarithm functions

Abstract

In 1934, Ramaswami proved a number of curious translation formulae satisfied by the Riemann zeta function. Such translation formulae, in turn give the meromorphic extension of the Riemann zeta function. In 1954, Apostol extended those identities to establish a family of such similar translation formulae. In this article, we establish many such Ramaswami and Apostol type translation formulae for the Dirichlet series defining the polylogarithm functions. This extended set up has many interesting applications, for example, it allows us to also find some (seemingly new) recurrence relations between the Bernoulli numbers, and use them to deduce some congruence properties of the tangent numbers.