= Shifted-exponential Poisson count recursion
{title2=$n(\beta+1)p_n=\{n+\alpha(\beta+1)\}p_{n-1}-\alpha p_{n-2}$}
For the <shifted-exponential Poisson mixture>, the displayed relation holds for $n\ge2$, 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.
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