Anyone recognize this single parameter discrete probability distribution?

[jacobcdf]

I have a single parameter discrete probability distribution defined over the domain of non-negative integers with pmf in k of:

Pr(k;L) = \frac{L^{k}}{k! * k! * I_{0}(2*\sqrt{L})}

Where I_{0}() is the modified Bessel function of the first kind with order 0.

I do know that E(k^{2}) = L.

Can anyone come up with a closed form for the distribution mean?

Does anyone recognize this distribution?

Thanks in advance,
J.

 

[jacobcdf]

I also know that as L \rightarrow \infty:

\gamma_{1} \rightarrow \sqrt{\frac{1}{2*\sqrt{K}}

and

\gamma_{2} \rightarrow \gamma_{1}^{2}

and E(k) appears to approach something approximated by:

\sqrt{L - \frac{\sqrt{L}}{2}}

But regardless, I still would like an exact closed-form solution, as the asymptotic approximation appears of little use practically.


J.

 

[jacobcdf]

In case anyone's interested, this distribution appears to be a special case of the Conway–Maxwell–Poisson distribution with the non-Poisson parameter \nu = 2.