Derivatives of the Hurwitz Zeta function for rational arguments

Abstract

The functional equation for the Hurwitz Zeta function ζ( s, a) is used to obtain formulas for derivatives of ζ( s, a) at negative odd s and rational a. For several of these rational arguments, closed-form expressions are given in terms of simpler transcendental functions, like the logarithm, the polygamma function, and the Riemann Zeta function.