Hurwitz Zeta Function - Series Representation

Series Representation

A convergent series representation defined for q > −1 and any complex s ≠ 1 was given by Helmut Hasse in 1930:

$\zeta(s,q)=\frac{1}{s-1} \sum_{n=0}^\infty \frac{1}{n+1} \sum_{k=0}^n (-1)^k {n \choose k} (q+k)^{1-s}.$

This series converges uniformly on compact subsets of the s-plane to an entire function. The inner sum may be understood to be the nth forward difference of ; that is,

where Δ is the forward difference operator. Thus, one may write

$\begin{align} \zeta(s, q) &= \frac{1}{s-1}\sum_{n=0}^\infty \frac{(-1)^n}{n+1} \Delta^n q^{1-s}\\ &= \frac{1}{s-1} {\log(1 + \Delta) \over \Delta} q^{1-s} \end{align}$