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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.2012-10-19
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    Lord_Farin, I will look after the polylogarithm function. Why do you say that the Poisson distribution is wrong?2012-10-19
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    $\lambda^n\exp(-\lambda)/n!$ is not of the form $C(n_0+n)^{-\gamma}$.2012-10-19
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    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.2012-10-19

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