Abstract
Recently, the partially degenerate Bell polynomials and numbers, which are a degenerate version of Bell polynomials and numbers, were introduced. In this paper, we consider the new type degenerate Bell polynomials and numbers, and obtain several expressions and identities on those polynomials and numbers. In more detail, we obtain an expression involving the Stirling numbers of the second kind and the generalized falling factorial sequences, Dobinski type formulas, an expression connected with the Stirling numbers of the first and second kinds, and an expression involving the Stirling polynomials of the second kind.
Highlights
Studies on degenerate versions of some special polynomials can be traced back at least as early as the paper by Carlitz [1] on degenerate Bernoulli and degenerate Euler polynomials and numbers.In recent years, many mathematicians have drawn their attention in investigating various degenerate versions of quite a few special polynomials and numbers and discovered some interesting results on them [2,3,4,5,6,7,8,9]
The partially degenerate Bell polynomials β n,λ ( x ), which are a degenerate version of Bell polynomials, were introduced (see Equation (3)) and some interesting identities on them were obtained in connection with Stirling numbers of the first and second kinds [14]
We study the new type degenerate Bell numbers and polynomials and we give some new identities for those polynomials and numbers
Summary
Studies on degenerate versions of some special polynomials can be traced back at least as early as the paper by Carlitz [1] on degenerate Bernoulli and degenerate Euler polynomials and numbers. The partially degenerate Bell polynomials β n,λ ( x ), which are a degenerate version of Bell polynomials, were introduced (see Equation (3)) and some interesting identities on them were obtained in connection with Stirling numbers of the first and second kinds [14]. In [14], the partially degenerate Bell polynomials are defined by the generating function to be e x(eλ (t)−1) =. From Equation (4), we can derive the generating function for the Stirling polynomials of the second kind given by (see [3,8,9,21,22]). In [7], Kim considered the degenerate Stirling polynomials of the second kind given by tn eλ (t) − 1)k eλx (t) = ∑ S2,λ (n, k | x ). We study the new type degenerate Bell numbers and polynomials and we give some new identities for those polynomials and numbers
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