A slew of mixture relationships involving discrete and continuous generalized hypergeometric distributions and their special cases

Our starting point is recognition of some mixture relationships involving the (continuous) Gauss hypergeometric distribution. Our main emphasis is then to generalize these relationships to ones involving (discrete) generalized hypergeometric distributions and their rarely considered continuous counterparts. Two such sets of relationships are derived one involving beta distributions and the other gamma distributions. A wide variety of interesting special cases arise along the way: Poisson, binomial, negative binomial, logarithmic, Conway–Maxwell–Poisson and Libby–Novick distributions all appear. There are also comments on the wider context within which the relationships of interest in this article arise.