User:BeyondNormality/Bhattacharya-negative binomial distribution

Also known as the generalized negative binomial, the probability mass function of the Bhattacharya-negative binomial distribution is given by

$$ \begin{align} \mathrm{BhattacharyaNB}(x;a,b,k) \equiv \Pr(X = x) & = \left(b^a (1+b)^{-a+k} (2+b)^{-k-x}\right) \sum _{j=0}^{\infty } \left(\frac{1}{2+b}\right)^j \binom{-1+a+j}{j} \binom{-1+j+k+x}{x} \\ & = b^a (b+1)^{k-a} (b+2)^{-k-x} \binom{k+x-1}{x} \, _2F_1\left(a,k+x;k;\frac{1}{b+2}\right) \end{align}$$
 * $$ x = 0,1,2, \dots $$
 * $$ a,k \ge 0 $$
 * $$ b > 0 $$

Expected Value
 * $$ \mathbb E(X) = \frac{a+b k}{b^2+b} $$

Variance
 * $$ \operatorname{Var}(X) = \frac{a (b (b+3)+1)+(b+2) b^2 k}{b^2 (b+1)^2} $$

Recurrence relation
 * $$ -\Pr (x-1) (a+b (k+2 x-2)+k+3 x-3)+(b+1) (b+2) x \Pr (x)+(k+x-2) \Pr (x-2)=0 $$

Moment Generating Function
 * $$ M_X(t)=\left(1-\frac{e^t-1}{b}\right)^{-a} \left(1-\frac{e^t-1}{b+1}\right)^{a-k} $$

Characteristic Function
 * $$ \varphi_X(t)=\left(1-\frac{e^{i t}-1}{b}\right)^{-a} \left(1-\frac{e^{i t}-1}{b+1}\right)^{a-k} $$

Probability Generating Function

G(t) = \left(1-\frac{t-1}{b}\right)^{-a} \left(1-\frac{t-1}{b+1}\right)^{a-k} $$