Two moments suffice for Bessel approximation: conditioned differences, contingency tables, and probabilistic divide-and-conquer

We develop Stein's method, with explicit constants throughout, for the discrete Bessel distribution: the law of a Poisson variable given its difference from an independent Poisson variable. The Bessel family is characterized by a birth-death operator with quadratic death rate $n(n+k)$ and as the exa