Identities for the Hurwitz zeta function, Gamma function, and L-functions

TL;DR

This paper derives new identities involving the Hurwitz zeta, Gamma, and Dirichlet L-functions, providing practical expansions for high-precision computation and formulas for special values.

Contribution

It introduces identities involving a polynomial sequence $eta_k(s)$ for the Hurwitz and related functions, extending previous work and generalizing Hasse's formula.

Findings

Practical expansions for high-precision evaluation

New identities involving special functions

Generalization of Hasse's formula

Abstract

We derive several identities for the Hurwitz and Riemann zeta functions, the Gamma function, and Dirichlet $L$-functions. They involve a sequence of polynomials $\alpha_k(s)$ whose study was initiated in an earlier paper. The expansions given here are practical and can be used for the high precision evaluation of these functions, and for deriving formulas for special values. We also present a summation formula and use it to generalize a formula of Hasse.