On univariate slash distributions, continuous and discrete

In this article, I explore in a unified manner the structure of uniform slash and $$\alpha $$α-slash distributions which, in the continuous case, are defined to be the distributions of Y / U and $$ Y_\alpha /U^{1/\alpha }$$Yα/U1/α where Y and $$Y_\alpha $$Yα follow any distribution on $$\mathbb {R}^+$$R+ and, independently, U is uniform on (0, 1). The parallels with the monotone and $$\alpha $$α-monotone distributions of $$ Y \times U$$Y×U and $$Y_\alpha \times U^{1/\alpha }$$Yα×U1/α, respectively, are striking. I also introduce discrete uniform slash and $$\alpha $$α-slash distributions which arise from a notion of negative binomial thinning/fattening. Their specification, although apparently rather different from the continuous case, seems to be a good one because of the close way in which their properties mimic those of the continuous case.