Some Identities of Degenerate Bell Polynomials

Taekyun Kim,Jongkyum Kwon,Dae San Kim,Han Young Kim

doi:10.3390/math8010040

Taekyun Kim, Jongkyum Kwon + Show 2 more

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https://doi.org/10.3390/math8010040

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Abstract

The new type degenerate of Bell polynomials and numbers were recently introduced, which are a degenerate version of Bell polynomials and numbers and are different from the previously introduced partially degenerate Bell polynomials and numbers. Several expressions and identities on those polynomials and numbers were obtained. In this paper, as a further investigation of the new type degenerate Bell polynomials, we derive several identities involving those degenerate Bell polynomials, Stirling numbers of the second kind and Carlitz’s degenerate Bernoulli or degenerate Euler polynomials. In addition, we obtain an identity connecting the degenerate Bell polynomials, Cauchy polynomials, Bernoulli numbers, Stirling numbers of the second kind and degenerate Stirling numbers of the second kind.

Highlights

For any nonzero λ ∈ R, the Carlitz’s degenerate Bernoulli polynomials are defined by t x λ (1 + λt) − 1 (1 + λt) λ = ∞ tn ∑ βn,λ (x) n!, (1) n =0When x = 0, β n,λ = β n,λ (0) are called the degenerate Bernoulli numbers.Note that limλ→0 β n,λ ( x ) = Bn ( x ), (n ≥ 0)
In our previous works related to this paper, we studied various degenerate versions of many special polynomials
They have been investigated by using several different means, such as generating functions, combinatorial methods, umbral calculus techniques, probability theory, p-adic analysis, differential equations, and so on

Summary

Introduction

For any nonzero λ ∈ R, the Carlitz’s degenerate Bernoulli polynomials are defined by (see [1]). The new type degenerate Bell polynomials are introduced by the generating function as (see [10]). In the recent paper [10], the new type degenerate Bell polynomials Beln,λ ( x ) (see (13)) were introduced and some interesting results about them were obtained, which are different from the previously defined partially degenerate Bell polynomials (see [9]) and a degenerate version of the ordinary Bell polynomials Beln ( x ) (see (8)). As a further study of the new type degenerate Bell polynomials, we will obtain two expressions involving these degenerate Bell polynomials, Carlitz’s degenerate Bernoulli polynomials and the Stirling numbers of the second kind, two identities involving those degenerate Bell polynomials, degenerate Euler polynomials and the Stirling numbers of the second kind. We will be able to find an identity involving those degenerate Bell polynomials, Cauchy polynomials, Bernoulli numbers, Stirling numbers of the second kind and degenerate Stirling numbers of the second kind

Some Identities of Degenerate Bell Polynomials

Conclusions