What is the sum of the following series? Is it some extended Zeta function?

Question

What is the sum of the following series? $$\sum_{m=0}^{\infty}\frac{(\nu)_m}{m!}\frac{1}{(m+1)^s}$$

where $(\nu)_m=\frac{\Gamma(\nu+m)}{\Gamma(\nu)}=\nu\cdot(\nu+1)\cdots(\nu+n-1))$ is the Pochhammer symbol.

Answer 1

I am unable to find the general solution of $$A_s=\sum _{m=0}^{\infty } \frac{ \Gamma (m+\nu )}{m! \,\Gamma (\nu )}\frac 1{(m+1)^{s}}$$ but it is almost sure that polylogarithm functions would be involved.

Using a CAS, what I got is $$A_1=\frac{1}{1-\nu }$$ $$A_2=\frac{\psi ^{(0)}(2-\nu )+\gamma }{1-\nu}$$ $$A_3=\frac{6 \psi ^{(0)}(2-\nu )^2+12 \gamma \psi ^{(0)}(2-\nu )-6 \psi ^{(1)}(2-\nu )+\pi ^2+6 \gamma ^2}{12 (1-\nu)}$$ and, for $s>3$, the expressions start to be really nasty.