# A generalization of gamma function

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 $$\Gamma^{\alpha}$$ ?

For $$\Re(\alpha) <-1,w > 0$$ then $$F_{\alpha,w}(s)=\int_0^\infty t^{-\alpha} (\frac{1-e^{-t}}{t})^{-\alpha} e^{-wt}t^{s-1}dt$$ is analytic for $$\Re(s) \ge 0$$.

For most $$\alpha,w$$, $$F_{\alpha,w}(0) \ne 0$$ and $$\zeta^\alpha(s,w)=\frac{F_{\alpha,w}(s)}{\Gamma(s)}$$ has a pole at $$s=0$$ so it doesn't make sense to look at its derivative.

The same holds for $$\Re(\alpha)\ge -1$$ except we need to continue the integral analytically from $$\zeta^\alpha(s,w) - \sum_{m=0}^M c_m(\alpha,w) \frac{1}{\Gamma(s)(s+m)}= \int_0^\infty t^{-\alpha} ( (\frac{1-e^{-t}}{t})^{-\alpha} e^{-wt} - \sum_{m=0}^M c_m(\alpha,w) t^m 1_{t < 1}) t^{s-1}dt$$ where $$(\frac{1-e^{-t}}{t})^{-\alpha} e^{-wt} = \sum_{m \ge 0}c_m(\alpha,w) t^m$$ and $$M > \Re(\alpha)$$.

$$(\frac{1-e^{-t}}{t})^{-\alpha} e^{-wt}$$ is analytic and rapidly decreasing near $$[0,\infty)$$ thus for $$\Re(s) > 0$$ $$\int_I (\frac{1-e^{-t}}{t})^{-\alpha} e^{-wt}t^{s-1}dt$$ $$= \int_0^\infty (\frac{1-e^{-t}}{t})^{-\alpha} e^{-wt}t^{s-1}dt-\int_0^\infty (\frac{1-e^{-t}}{t})^{-\alpha} e^{-wt}(e^{-2i\pi}t)^{s-1}dt$$ $$= (1-e^{-2i \pi (s-1)})\int_0^\infty (\frac{1-e^{-t}}{t})^{-\alpha} e^{-wt}t^{s-1}dt$$

where $$I$$ is the contour $$\infty \to \infty$$ enclosing positively $$[0,\infty)$$.

The contour integral $$\int_I (\frac{1-e^{-t}}{t})^{-\alpha} e^{-wt}t^{s-1}dt$$ is entire in $$s$$.

For $$\Re(s) > |\alpha|$$ by absolute convergence $$\int_0^\infty (1-e^{-t})^{-\alpha} e^{-wt}t^{s-1}dt= \sum_{n=0}^\infty {-\alpha \choose n} \int_0^\infty e^{-(w+n)t} t^{s-1}dt=\sum_{n=0}^\infty {-\alpha \choose n} (w+n)^{-s}\Gamma(s)$$