Let's define discrete-analytic functions as functions that are equal to their Newton series expansion:

$$f(x) = \sum_{k=0}^\infty \binom{x-a}k \Delta^k f(a)$$

My question is whether $\zeta(s,q)$ ($q$=const) is discrete-analytic against $s$?

That is whether its Newton series converges and is equal to the function itself.

For comparison, in the following graphic there are four functions:

**red**is the function $\zeta(x,3)$**blue**is $\frac{\cos (\pi x)\psi_b^{(x+1)}(3)}{\Gamma(x+2)}$ where $\psi_b$ is the balanced polygamma**yellow**is $\frac{\cos (\pi x)\psi^{(x+1)}(3)}{\Gamma(x+2)}$ where $\psi$ is the polygamma as implemented in Mathematica**green**is the partial Newton expansion of the above functions taken at first 20 terms.

The three first functions and the Newton expansion, if it converges, have the same values at non-negative integer arguments.

**notation**
$\zeta(x,q)$ is the Hurwitz zeta function, LINK

Balanced polygamma LINK