On the gamma difference distribution

Abstract

The gamma difference distribution is defined as the difference of two independent gamma distributions, with in general different shape and rate parameters. Starting with knowledge of the corresponding characteristic function, a second order linear differential equation specification of the probability density function is given. This is used to derive a Stein-type differential identity relating to the expectation with respect to the gamma difference distribution of a general twice differentiable function g(x). Choosing g(x)=xk gives a second order recurrence for the positive integer moments, which are also shown to permit evaluations in terms of 2F1 hypergeometric polynomials. A hypergeometri..