Ramaswami Type translation formulae for the polylogarithm functions

TL;DR

This paper extends Ramaswami and Apostol's translation formulae to polylogarithm functions, revealing new recurrence relations for Bernoulli numbers and congruences for tangent numbers, enriching the understanding of special functions.

Contribution

It introduces new translation formulae for polylogarithms, generalizing previous identities for the Riemann zeta function and deriving novel properties of Bernoulli and tangent numbers.

Findings

Derived new recurrence relations for Bernoulli numbers.

Established congruence properties of tangent numbers.

Extended translation formulae to 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.