Multivariate Pascal Polynomials of Order K with Probability Applications

Abstract

For any fixed positive integer k, let C _k(n, m) be the entry at the intersection of row n (n ≥ 0) and column m (m ≥ 0) in the Pascal triangle of order k, viz., T _k. Then

$$C_{1}(n,0)=1 \text{ for} n \geq 0 \text{ and} C_{1}(n,m)=0 \text{ for} m\geq 1$$

, and for k ≥ 2, C _k(0,0) = 1, C _k(0,m) = 0 for m ≥ 1, and

$$ {{C}_{k}}(n,m) = \left\{ {\begin{array}{*{20}{c}} {\sum\limits_{{i = 1}}^{m} {{{C}_{k}}} (n - 1,m - i), 0 \leqslant m \leqslant k - 1{\text{ and}} n \geqslant 1} \hfill \\ {\sum\limits_{{i = 0}}^{{k - 1}} {{{C}_{k}}} (n - 1,m - i), m \geqslant k{\text{ and}} n \geqslant 1.} \hfill \\ \end{array} } \right. $$

(1)

.