Shifted-exponential Poisson count recursion ($n(\beta +1)p_n=\{n+\alpha (\beta +1)\}{p}_{n-1}-\alpha p_{n-2}$)

For the shifted-exponential Poisson mixture, the displayed relation holds for $n\ge 2$, initialized by $p_0=e^{-\alpha }\beta /(\beta +1)$ and $p_1=(\alpha +1/(\beta +1))p_0$. It follows by differentiating the probability generating function and comparing coefficients. Nonnegativity follows from the independent Poisson-geometric convolution of independent random variables representation.