Expansion of Dirichlet L-function on the critical line in Meixner-Pollaczek polynomials

Abstract

We study the expansions of the completed Riemann zeta function and completed Dirichlet L-functions in Meixner-Pollaczek polynomials. We give the proof of the uniform convergence, the multiplicative structure for the coefficients of these expansions, and a calculation for the coefficients of a completed Dirichlet L-function $\hat{L}(s, \chi_{-1})$. Furthermore, we give a boundary for the zeros of the approximating polynomial.