On Generalized Bell Polynomials

Abstract

It is shown that the sequence of the generalized Bell polynomialsSn(x)is convex under some restrictions of the parameters involved. A kind of recurrence relation forSn(x)is established, and some numbers related to the generalized Bell numbers and their properties are investigated.

A combinatorial analysis of higher order generalised geometric polynomials: A generalisation of barred preferential arrangements

“…This is a generalisation of the work done in [6]. (1−x(e t −1)) r , for instance in [7,10,11,12,14,15,16,17,18,21]. Hence, our results in Theorem 3.1 offers a generalised combinatorial interpretation of these geometric polynomials.…”

“…Part II of [9]). These polynomials are well known in the literature, and it is also well known that these polynomials arise from variations of the generating function e rt (1−x(e t −1)) r , for instance in [7,10,11,12,14,15,16,17,18,21]. In this study our combinatorial interpretation of the integer sequences arising from the generating function…”

“…For the case λ = 1 the statement of Theorem 3.1 can be derive from the one given for the numbers B n (α, β, γ) in [7].…”

Differential Equations Arising from the Generating Function of the (r, β)-Bell Polynomials and Distribution of Zeros of Equations

“…In particular, the generalized Bell polynomials B n (x, −λ) = E λ [(Z + x − λ) n ], λ, x ∈ R, n ∈ N, where Z is a Poission random variable with parameter λ > 0 (see [1][2][3]). The (r, β)-Bell polynomials G n (x, r, β) are this formula using the generating function:…”

The estimation of the zeros of the Bell and r-Bell polynomials

“…In this paper we study the structure of zeros of the Bell and r-Bell polynomials. Estimations of these polynomials and the r-Stirling numbers are well known [6][7][8]. What now interest us is the leftmost zero of these polynomials.…”