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The actual textbook question:

An integer valued random variable $X$ has p.m.f. $f$ satisfying $$f(x)=\frac{\alpha+\beta x}{x}f(x-1)\quad...(*),\quad\beta \ne1\quad\text{and}\quad x=1,2,...$$Find $\mu,\mathrm{Var}(X)$ and $\mathrm{MD}_{\mu}$ where $\mu=\mathrm{E}(X)$.

While this problem was solvable by cross-multiplying the recursive equation of $f$ and summing up both sides for all possible $x$ to find the required moments, I wasn't able to find out the exact closed-form expression for $f$. I realise that we don't need the exact expression to find the required quantities, but is it possible to solve for $f$ from the given relation? The source adds that the binomial and Poisson random variables are of this type.

Putting the values of $x$ in $(*)$ for a finite number of times from $x=1$ to $x=n$ and multiplying the $n$ equations I get, $$f(n)=\frac{f(0)}{n!}(\alpha+\beta)(a+2\beta)...(a+n\beta)$$

Then I could perhaps write $\displaystyle f(x)=\lim_{n\to\infty}f(n)=\frac{f(0)}{n!}\lim_{n\to\infty}p_n$, (say).

Applying the AM-GM inequality on $p_n$, I could again possibly say that $$\lim_{n\to\infty}p_n=\lim_{n\to\infty}\left[\alpha+\frac{\beta (n+1)}{2}\right]^n,\quad\text{which diverges}.$$

I don't think using $\sum_{x=1}^{\infty}f(x)=1$ helps me out here.

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    In the actuarial literature, using your notation, these are known as the [$(\beta, \alpha, 0)$ class of distributions](https://en.wikipedia.org/wiki/(a,b,0)_class_of_distributions). I would guess that there's probably a derivation of these quantities in *Loss Models* by Klugman and can check when I get home. – 2017-01-09
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    @Clarinetist Do you think this post is more suitable on Cross-validated? – 2017-01-09
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    I don't think it matters for this question; one could take this question as a probability question. – 2017-01-09
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    @Clarinetist If there is an advanced derivation involved then I think I have been trying in vain. – 2017-01-09
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    *Loss Models* is by no means an advanced text (about a junior/senior-level undergraduate text). – 2017-01-09
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    Some further research: this is also known as the [Panjer class](http://www.actuaries.org/ASTIN/Colloquia/Helsinki/Papers/S7_13_Fackler.pdf) of distributions; see Corollary 1 on p. 4. The [Panjer recursion](https://en.wikipedia.org/wiki/Panjer_recursion) is used to prove these formulas. [This](https://matthewhr.wordpress.com/2012/12/12/panjer-recursion/) goes through how to derive the probability generating function, from which you could calculate moments. – 2017-01-09
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    [These sample pages](http://www.springer.com/cda/content/document/cda_downloaddocument/9783540928997-c2.pdf?SGWID=0-0-45-714009-p173875410) discuss the derivation in detail on p. 42. – 2017-01-09
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    @Clarinetist Thanks for that. – 2017-01-09

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