On the gamma difference distribution

TL;DR

This paper characterizes the gamma difference distribution, deriving its probability density function, moments, and related identities using differential equations and hypergeometric functions, and connects it to the variance gamma distribution.

Contribution

It provides a differential equation characterization and moment formulas for the gamma difference distribution, including hypergeometric evaluations, extending previous results for the variance gamma distribution.

Findings

Derived a second order differential equation for the PDF.

Established recurrence relations for moments.

Connected the distribution to hypergeometric functions.

Abstract

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.