A new expansion of the Riemann zeta function

Abstract

This article presents polynomial expansions for the Dirichlet eta function and Riemann zeta function that are convergent in the critical strip. To do this we introduce a family of hypergeometric polynomials, whose roots lie on the line $\{\Re(s)=1/2\}$, and that are related to Meixner-Pollaczek polynomials. We also obtain orthonormal expansions for eta and zeta restricted to the line $\{\Re(s)=1/2\}$. The coefficients of these expansions are given explicitly as linear combinations with rational coefficients of $\log(2),$ Euler's constant $γ$, and zeta values at positive integers.