A New Kind of Numbers Appearing in the n-Fold Convolution of Truncated Binomial and Negative Binomial Distributions

In the present paper the numbers $C(m,n,s)$ appearing in the n-fold convolution of truncated binomial and negative binomial distributions (see T. Cacoullos and Ch. Charalambides [1]) are introduced by \[ C(m,n,s) = \frac{1}{{n!}}[\Delta ^n (sx)_m ]_{x = 0}, \quad m,n\,{\text{positive integers and}}\,s\,{\text{a real number}} \] where $\Delta $ denotes the difference operator and $(sx)_m = sx(sx - 1) \cdots (sx - m + 1)$ is the usual falling factorial. This definition is shown to be equivalent to that given in [1]. A recurrence relation for these numbers, useful for tabulation purposes, is obtained. The difference equation is solved by using the exponential generating function of the numbers $C(m,n,s)$; a third expression of these numbers is concluded. Relations between Stirling, Lah and C-numbers and limiting expressions containing these numbers are also derived. Finally, some applications of the numbers $C(m,n,s)$ in probability theory and in the expansion of a composite function are given.