The pmf of the Borel distribution is given by
$$f(x)=\frac{e^{-x\rho } (x \rho )^{x-1}}{x!},\ \{x \in 1,2,\dots\}$$
I'm trying to compute the cdf, which is
$$F(x) = \sum_{j=1}^x \frac{e^{-j \rho } (j \rho )^{j-1}}{j!}.$$
Does anyone know of a closed form for the cdf of the Borel distribution?
========== UPDATE ====================
The closest I could find was this:
Haight, F. (1961). A Distribution Analogous to the Borel-Tanner. Biometrika, 48(1/2), 167-173.
In there, they have...
$$B(y,r)=\frac{r}{y} \binom{2 y-r-1}{y-1}$$
$$f(y)=B(y,r)\frac{\alpha ^{y-r}}{(\alpha +1)^{2 y-r}},\ y\in\{r,\dots,\infty\}, r \geq 1, r\in Z,$$
which has does have a cdf in terms of Hypergeometric functions.
But this distribution is NOT THE SAME as the Borel. I.e., they have the same mean, but different moments. I am unclear what they mean by ``analagous".