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.

翻译：伽马差分分布定义为两个伽马分布之差，通常具有不同的形状参数和速率参数。基于相应特征函数的知识，给出了概率密度函数的二阶线性微分方程表征。由此推导出与一般二阶可微函数 $g(x)$ 关于伽马差分分布的期望相关的Stein型微分恒等式。选取 $g(x) = x^k$ 可得到正整数阶矩的二阶递推关系，这些矩还可表示为 ${}_2 F_1$ 超几何多项式。给出了绝对连续矩的超几何函数表达式。将伽马差分分布特殊化可得到方差伽马分布。本文所得类型的结果此前曾针对该分布获得，从而可进行比较分析。