Abstract
The aim of this paper is to study the degenerate Bell numbers and polynomials which are degenerate versions of the Bell numbers and polynomials. We derive some new identities and properties of those numbers and polynomials that are associated with the degenerate Stirling numbers of both kinds.
Highlights
Introduction e Bell number Beln counts the number of partitions of a set with n elements into disjoint nonempty subsets. e Bell polynomials Beln(x), called Touchard or exponential polynomials, are natural extensions of Bell numbers. e partial and complete Bell polynomials, which are multivariate generalizations of the Bell polynomials, have diverse applications in mathematics and in physics and engineering as well
We derive several properties and identities of those numbers and polynomials which include recurrence relations for degenerate Bell polynomials, and expressions for them that can be derived from repeated applications of certain operators to the exponential functions, the derivatives of them (Corollary 1), the antiderivatives of them, and some identities involving them
We studied by means of generating functions the degenerate Bell polynomials which are degenerate versions of the Bell polynomials
Summary
Us, by comparing the coefficients on both sides of (31) and from (28), we obtain the following theorem. Erefore, by (40) and eorem 2, we obtain the following theorem. For n ≥ 0, the following identity holds: Beln+1,λ(x) xBeln′,λ (x) + Beln,λ(x) − nλBeln,λ(x), (41). Erefore, by (49), we obtain the following corollary. Complexity erefore, by (55) and (56), we obtain the following theorem. Erefore, by comparing the coefficients on both sides of (73), we obtain the following theorem. Erefore, by (76) and (79), we obtain the following theorem. N ≥ 0, the following identity holds true.
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