Entropy of bivariate negative binomial distribution

Solution 1:

In your reference [1], it says that the marginal distribution for $\mathsf{X}+\mathsf{Y}$ is equally simple, at the bottom of page 79. The parameters that Dunn uses for the NBin and the bivariate NBin are such that $\mathsf{X}+\mathsf{Y}$ is negative binomial with $p=p_1+p_2$ and $A=a$. Note in reference [2] Cheraghchi calls $A$ by $r$. Then you should be able to calculate that by using the 2nd-to-last displayed equation in [2] page 14. Take $\alpha=a$ and note that $(x+y+a-1)!$ is $\Gamma(x+y+a)$. So in that formula, that $\alpha$ should be $a$ and also that $r$ should be $a$. And $p$ should be $p_1+p_2$.