I want to know if there is a "closed form" of the following generating function,
$G_n(x) = \sum_{n=0}^{\infty} P_n x^n$
where,
$P_n = C(n_0 + n)^{-\gamma}$
where $C$ is a normalization constant, and where $n_0 \simeq 1$ and $2 \leq \gamma \leq 3$. By closed form I mean a function in terms of some classical or special function. For example, if $P_m$ is a Poisson distribution,
$P_m = \frac{\lambda^m e^{-\lambda}}{m!}$
then its generating function $G_m(x)$ can be written in closed form as,
$G_m(x) = e^{\lambda(x-1)}$
Best Regards !!!
Juan