Derivatives of the Hurwitz zeta function for rational arguments (Q1298658)

A classic functional equation expresses the Hurwitz zeta function \(\zeta(s,p/q)\) for a positive rational \(p/q\), where \(p<q\), as a finite linear combination of terms of the form \(\zeta(1-s,n/q)\) with trigonometric coefficients. The authors differentiate this relation and then take \(s=-2k+1\), where \(k\) is a positive integer. This gives a closed form evaluation of \(\zeta'(-2k+1,p/q)\) as a linear combination of terms of the form \(\zeta'(2k,n/q)\) together with a linear combination of polygamma functions \(\psi^{(2k-2)} (n/q)\), plus terms involving Bernoulli polynomials evaluated at rational values. In special cases such as \(p/q=1/2, 1/3, 1/4, 1/6, 2/3, 3/4\), and \(5/6\) the formulas simplify considerably.