(Received April 24, 1989; revised February 13, 1990)
Abstract. The distribution with probability function pk(n, alpha, beta) = An,k(alpha, beta)/(alpha + beta)[n], k = 0, 1, 2,...., n, where the parameters alpha and beta are positive real numbers, An,k(alpha, beta) is the generalized Eulerian number and (alpha + beta)[n] = (alpha + beta)(alpha + beta + 1)···(alpha + beta + n - 1), introduced and discussed by Janardan (1988, Ann. Inst. Statist. Math., 40, 439-450), is further studied. The probability generating function of the generalized Eulerian distribution is expressed by a generalized Eulerian polynomial which, when expanded suitably, provides the factorial moments in closed form in terms of non-central Stirling numbers. Further, it is shown that the generalized Eulerian distribution is unimodal and asymptotically normal.
Key words and phrases: Eulerian numbers, Eulerian polynomials, Stirling numbers, random permutations, unimodality, asymptotic normality.