Fast formulas for the Hurwitz values $\zeta(2,a)$ and $\zeta(3,a)$

Abstract

We prove two fast formulas for the Hurwitz values $\zeta(2,a)$ and $\zeta(3,a)$ respectively with the help of the WZ method. In them $(a)_n$ denotes the rising factorial or Pochhammer's symbol defined by $(a)_0=1$ and $(a)_n=a(a+1)\cdots(a+n-1)$ for positive integers $n$. The Huwitz $\zeta$ function is defined by $\zeta(s,a)=\zeta(0,s,a)=\sum_{k=0}^{\infty} (k+a)^{-s}$. In addition, we can use these fast evaluations to compute also in a rapid way Dirichlet values of the kinds $L_{\chi}(2)$ and $L_{\chi}(3)$.