how to prove that $\zeta(s,q)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt$

Question

how to prove that $$\zeta(s,q)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt:R(s)>1 , R(q)>0$$

where $\zeta(s,q)$ is Hurwitz zeta function

Answer 1

Note that $$\int_{0}^{\infty}\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt=\sum_{k=0}^\infty\int_{0}^{\infty}t^{s-1}e^{-(q+k)t}dt$$

For each term of the series we have $$\int_{0}^{\infty}t^{s-1}e^{-(q+k)t}dt= \frac{1}{(q+k)^{s-1}}\int_{0}^{\infty}x^{s-1}e^{-x}\frac{dx}{q+k}=\frac{\Gamma(s)}{(q+k)^{s}}$$

Adding up we get the series representation of the Hurwiz zeta function $$\zeta(s,q)=\sum_{k=0}^\infty\frac{1}{(q+k)^s}$$