Abstract
Carlitz initiated a study of degenerate Bernoulli and Euler numbers and polynomials which is the pioneering work on degenerate versions of special numbers and polynomials. In recent years, studying degenerate versions regained lively interest of some mathematicians. The purpose of this paper is to study degenerate Bell polynomials by using umbral calculus and generating functions. We derive several properties of the degenerate Bell polynomials including recurrence relations, Dobinski-type formula, and derivatives. In addition, we represent various known families of polynomials such as Euler polynomials, modified degenerate poly-Bernoulli polynomials, degenerate Bernoulli polynomials of the second kind, and falling factorials in terms of degenerate Bell polynomials and vice versa.
Highlights
Introduction and preliminariesIn [3, 4], Carlitz studied degenerate Bernoulli and Euler polynomials, which are degenerate versions of the ordinary Bernoulli and Euler polynomials, and investigated some combinatorial results as well as some arithmetical ones
We have witnessed in recent years that, along the same line as Carlitz’s pioneering work, some mathematicians began to explore degenerate versions of quite a few special polynomials and numbers which include the degenerate Bernoulli numbers of the second kind, degenerate Stirling numbers of the first and second kinds, degenerate Cauchy numbers, degenerate Bell numbers and polynomials, degenerate complete Bell polynomials and numbers, degenerate gamma function, and so on
After finding a formula expressing any polynomial in terms of the degenerate Bell polynomials, we applied this formula to Euler polynomials and powers of x
Summary
The Bell number Bn counts the number of partitions of a set with n elements into disjoint nonempty subsets. The purpose of the present paper is to study the degenerate Bell polynomials and numbers by means of umbral calculus and generating functions. The degenerate Bernoulli polynomials of the second kind of order r are defined by t logλ(1 + t). For x = 0, b(nr,)λ = b(nr,)λ(0) are called the degenerate Bernoulli numbers of the second kind of order r. As an inversion formula of (12), the Stirling numbers of the second kind are defined as n xn = S2(n, l)(x)l (n ≥ 0), (see [1, 16, 21]). In [9], the degenerate Stirling numbers of the first kind are defined by Kim and Kim as follows: n (x)n = S1,λ(n, l)(x)l,λ (n ≥ 0).
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