Identities for the Riemann zeta function

TL;DR

This paper introduces new polynomial-based expansions for the Riemann zeta function, extending known series and offering a novel approach to its analytic continuation.

Contribution

It presents generalized polynomial identities for $ta(s)$, extending existing series expansions and providing a new method for analytic continuation.

Findings

New polynomial expansions for $ta(s)$ involving generalized Stirling numbers.

Extensions of known series expansions to complex values of $s$.

A different approach to the analytic continuation of the zeta function.

Abstract

We obtain several expansions for $\zeta(s)$ involving a sequence of polynomials in $s$, denoted in this paper by $\alpha_k(s)$. These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities extend some series expansions for the zeta function that are known for integer values of $s$. The expansions also give a different approach to the analytic continuation of the Riemann zeta function.