**Sankhya: The Indian Journal of Statistics**

1995, Volume 57, Series A, Pt. 1, pp. 41--55

**ON A TWO-PARAMETER FAMILY OF DISCRETE UNIMODAL RANDOM VARIABLES**

By

EMILE BERTIN and RADU THEODORESCU, *Universiteit Utrecht and Université Laval*

SUMMARY. This paper studies a new class of **Z**-valued random variables, called beta unimodal, which are 'dot products'(in the sense of Steutel and van Harn, 1979) of the form $U \odot Z$, where U is beta distributed and independent of the **Z**-valued random variable Z. As a particular case, beta unimodality contains the version of $\alpha$-unimodality on **N**, described in Abouammoh (1987), Steutel (1988). The key result (Theorem 3.8) shows that the space of all beta unimodal probability distributions is isomorphic with the space of all probability measures on **Z**, entailing many similarities between beta unimodality on **Z** and classical unimodality on **R**.

*AMS (1990) subject classification.* Primary 60E05; secondary 26A33, 26B25.

*Key words and phrases.* Discrete unimodality, beta unimodality, generalized unimodality, monotonicity, fractional derivative, fractional difference, hypergeometric function.