On mixture relationships between central and non-central chi-squared difference distributions

Gaunt (2026) introduced and studied the non-central chi-squared difference distribution. One version of this arose as the distribution of the product of the two marginal random variables of the general (non-degenerate) bivariate normal distribution. Initially, I provide an alternative basic argument as to why this is so. The main focus of the article, however, is to extend a number of known mixing relationships between (non-central) chi-squared distributions to the (non-central) chi-squared difference distribution case. In certain circumstances, just a single mixing random variable remains required rather than the two independent ones that are trivially applicable in general.