# What is the closed form of generating function of a power law?

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

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You are aware that the Poisson distribution is not of the form that you introduced? However, have a look at the polylogarithm function. – Lord_Farin Oct 19 '12 at 13:42
Lord_Farin, I will look after the polylogarithm function. Why do you say that the Poisson distribution is wrong? – Juan I. Perotti Oct 19 '12 at 13:49
$\lambda^n\exp(-\lambda)/n!$ is not of the form $C(n_0+n)^{-\gamma}$. – Lord_Farin Oct 19 '12 at 15:12
ohh, now I understand. I just redefined $P_n$ it to exemplify by what I mean when I say "closed form". I will change the notation. – Juan I. Perotti Oct 19 '12 at 18:17