Fast formulas for the Hurwitz values $\hat{I}\P (2,a)$ and $\hat{I}\P (3,a)$

Abstract

We prove two fast formulas for the Hurwitz values $\hat{I}\P (2,a)$ and $\hat{I}\P (3,a)$ 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 $\hat{I}\P$ function is defined by $\hat{I}\P (s,a)=\hat{I}\P (0,s,a)={\sum }_{k=0}^{\infty }(k+a{)}^{-s}$. In addition, we can use these fast evaluations to compute also in a rapid way Dirichlet values of the kinds $L_{\ddot{I}}‡(2)$ and $L_{\ddot{I}}‡(3)$.

7 pages