For any prime p we establish a congruence of Kummer type for the NN orlund polynomial B (x) n , and determine the highest power of p which divides the coeecient denominators of the NN orlund polynomial. This improves the result of 4] where only an upper bound was proven. We deduce a simple formula for the least common denominator of the coeecients. Applications are made for the Stirling polynom...

In this paper, we define two natural (p, q)-analogues of the generalized Stirling numbers of the first and second kind S(α, β, r) and S(α, β, r) as introduced by Hsu and Shiue [17]. We show that in the case where β = 0 and α and r are nonnegative integers both of our (p, q)-analogues have natural interpretations in terms of rook theory and derive a number of generating functions for them. We al...

The normal ordering formulae for powers of the boson number operator n̂ are extended to deformed bosons. It is found that for the “M-type” deformed bosons, which satisfy aa− qaa = 1, the extension involves a set of deformed Stirling numbers which replace the Stirling numbers occurring in the conventional case. On the other hand, the deformed Stirling numbers which have to be introduced in the ca...

In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de…ned (q; r; w)Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q...

The left-truncated generalized Poisson distribution belongs to the family of the modified power series distributions. Using sufficiency and completeness of £ ^ x . (minx , , $ 3 x 0 ) respectively, when the truncation point is known (resp. unknown), the minimum variance unbiased (M.V.U) estimator for certain functions of the parameter $ (resp. 0, r) involved in these distributions can be obtain...

We show how generalized Zagreb indices $M_1^k(G)$ can be computed by using a simple graph polynomial and Stirling numbers of the second kind. In that way we explain and clarify the meaning of a triangle of numbers used to establish the same result in an earlier reference.

Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, k-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in [17]. Previous well-known Stirling functions introduced by Butzer and Hauss [4], Butzer, Kilbas, and Trujilloet [6] and others are included as particular cases of our generali...

For a graph G and a positive integer k, the graphical Stirling number S(G, k) is the number of partitions of the vertex set of G into k non-empty independent sets. Equivalently it is the number of proper colorings of G that use exactly k colors, with two colorings identified if they differ only on the names of the colors. If G is the empty graph on n vertices then S(G, k) reduces to S(n, k), th...

Abstract In this paper, we introduce a kind of combinatorial numbers, D − Stirling numbers, and its special cases. An exponential generating function of the D − Stirling numbers is given. We also present recurrence relations, monotonicity, and limiting properties of the D−Stirling numbers. Applications to statistical probability function estimation and restricted occupancy theory are provided. ...