For $\alpha\in\mathbb{C}$, I defined the "complex-weighted" Hurwitz zeta function \begin{eqnarray*}\displaystyle \zeta^{\alpha}(s,w)=\frac{1}{\Gamma(s)}\int_0^{\infty} \frac{e^{-wt}}{(1-e^{-t})^{\alpha}}t^{s-1}\,dt.\end{eqnarray*}

From this definition, we can consider a generalization of the gamma function \begin{eqnarray*}\displaystyle \Gamma^{\alpha}(w)=\exp\left(\left.\frac{\partial}{\partial s}\zeta^{\alpha}(s,w)\right|_{s=0}\right).\end{eqnarray*} In particular, $\Gamma^1(w)=\Gamma(w)/\sqrt{2\pi}$. Then I have three questions:

(1) The following representation is true? \begin{eqnarray*}\displaystyle \zeta^{\alpha}(s,w)=\sum_{n=0}^{\infty} \frac{(\alpha)_n}{n!}(n+w)^{-s},\end{eqnarray*} where $(\alpha)_n=\alpha\cdots(\alpha+n-1)$ is the Pochhammer symbol.

(2) If we could get a representation \begin{eqnarray*}\displaystyle \zeta^{\alpha}(s,w)=\frac{1}{\Gamma(s)(A(\alpha)e^{2\pi is}-1)}\int_{I(\lambda,\infty)} \frac{e^{-wt}}{(1-e^{-t})^{\alpha}}t^{s-1}\,dt,\end{eqnarray*} then what is $A(\alpha)$ ? $I(\lambda, \infty)$ is the path consisting of the infinite line from $\infty$ to $\lambda$, the circle of radius $\lambda$ around $0$ in the positive sense and the infinite line from $\lambda$ to $\infty$

(3) Is there any paper about $\zeta^{\alpha}$ or $\Gamma^{\alpha}$ ?