A special class of Bell polynomials

Abstract

We prove that the V(n, k) are also related to the numbers W(n, k) defined by fc! 2 W(n, k)x"/n< = [(*-2)(e*-l)]fc n=0 in much the same way the associated Stirling numbers are related to the Stirling numbers. Finally, we examine, more generally, the Bell polynomials Bn k(ax, a2, 3-a, 4-a, 5-a, ...) and show how the methods of this paper can be used to prove several formulas involving the Bernoulli and Stirling numbers.

Some Inequalities of the Bell Polynomials

“…Therefore, applications of the Bell polynomials B k (x) to integrable nonlinear equations are greatly expected and any amendment on multilinear forms of soliton equations, even on exact solutions, would be beneficial to interested audiences in the research community. For more information about the Bell polynomials B k (x), please refer to [6,7,18,19] and closely related references therein. In this paper, continuing the article [16], we present an explicit formula and its inversion formula for higher order derivatives with respect to t of generating functions e xe ±t for the Bell polynomials B k (x) with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind B n,k (x 1 , x 2 , .…”

Super extension of Bell polynomials with applications to supersymmetric equations

“…The Bell polynomials have been exploited in combinatorics, statistics and other fields [23]- [25]. Some generalized forms of Bell polynomials already appeared in literature [26]- [30]. More recently Lambert, Gilson et al found that the Bell polynomials also play important role in the characterization of bilinearizable equations.…”

New Bilinear Bäcklund Transformation and Lax Pair for the Supersymmetric Two‐Boson Equation

“…The Bell polynomials have been exploited in combinatorics, statistics, and other fields [15][16][17]. Some generalized forms of Bell polynomials have already appeared in literature [18][19][20][21][22].…”