  The gamma difference distribution is defined as the difference of two gamma
distributions, with in general different shape and rate parameters. Starting
with knowledge of the corresponding characteristic function, a second order
linear differential equation characterisation 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 $g(x) = x^k$ gives
a second order recurrence for the positive integer moments, which are also
shown to permit evaluations in terms of ${}_2 F_1$ hypergeometric polynomials.
A hypergeometric function evaluation is given for the absolute continuous
moments. Specialising the gamma difference distribution gives the variance
gamma distribution. Results of the type obtained herein have previously been
obtained for this distribution, allowing for comparisons to be made.
