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：伽马差分布定义为两个伽马分布的差值，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已知性质，本文给出了概率密度函数的二阶线性微分方程表征。利用该表征推导出关于一般二阶可微函数$g(x)$的伽马差分布期望的斯坦型微分恒等式。选择$g(x) = x^k$可得到正整数阶矩的二阶递推关系，且这些矩可表示为${}_2 F_1$超几何多项式。针对绝对连续矩给出了超几何函数求值方法。将伽马差分布特例化可得到方差伽马分布。此前针对该特例分布已获得类似形式的结果，便于进行对比分析。