User:BeyondNormality/Barton distribution

The probability mass function of the Barton distribution is



\mathrm{Barton}(x; K,N) \equiv \Pr(X = x) = \frac{1}{x!}\sum _{j=x}^N \frac{(-1)^{j-x} (K!)^j \left(N_{(j)}\right)^2}{(j-x)! (K N)_{(j K)}} $$


 * $$ x = 0, 1, \dots, N $$
 * $$ K \in \mathbb{N}  $$
 * $$ N \in \mathbb{N}_0  $$
 * $$ (n)_{(k)}=n(n-1)(n-2)\cdots(n-k+1) $$ (Falling Factorial)

Probability Generating Function



G(t) = \,_1F_{K-1}\left(-N; \frac{1}{K}-N, \frac{2}{K}-N, \dots, \frac{K-1}{K}-N; \frac{K! (t-1)}{{(-K)}^K} \right) $$