Degenerate Poly-Lah-Bell Polynomials and Numbers

Taekyun Kim,Hye Kyung Kim,Ali Jaballah

doi:10.1155/2022/2917943

Taekyun Kim, Hye Kyung Kim + Show 1 more

Open Access

https://doi.org/10.1155/2022/2917943

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Abstract

Many mathematicians studied “poly” as a generalization of the well-known special polynomials such as Bernoulli polynomials, Euler polynomials, Cauchy polynomials, and Genocchi polynomials. In this paper, we define the degenerate poly-Lah-Bell polynomials arising from the degenerate polyexponential functions which are reduced to degenerate Lah-Bell polynomials when k = 1 . In particular, we call these polynomials the “poly-Lah-Bell polynomials” when λ ⟶ 0 . We give their explicit expression, Dobinski-like formulas, and recurrence relation. In addition, we obtain various algebraic identities including Lah numbers, the degenerate Stirling numbers of the first and second kind, the degenerate poly-Bell polynomials, the degenerate poly-Bernoulli numbers, and the degenerate poly-Genocchi numbers.

Highlights

Many mathematicians have studied “poly” for the wellknown special polynomials such as poly-Bernoulli polynomials, poly-Euler polynomials, poly-Bell polynomials, and poly-Genocchi polynomials [1–10]
A lot of research has been done on degenerate versions of various special polynomials and numbers, such as Stirling numbers, Bernoulli polynomials, Euler polynomials, Genocchi polynomials, and Bell polynomials, which has given mathematicians a renewed interest in various degenerate polynomials and numbers [1, 3, 4, 6, 7, 9, 11]
We considered the degenerate poly-Lah-Bell polynomials and numbers by using of the degenerate polyexponential functions and obtained some combinatorial identities between these polynomials and numbers and special polynomials and numbers, involving the explicit formula, Dobinski-like formula, and recurrence relation

Summary

Introduction

Many mathematicians have studied “poly” for the wellknown special polynomials such as poly-Bernoulli polynomials, poly-Euler polynomials, poly-Bell polynomials, and poly-Genocchi polynomials [1–10]. In this paper, we define the degenerate polyLah-Bell polynomials by means of the degenerate polyexponential functions called the “poly-Lah-Bell polynomial”s when λ ⟶ 0. They are reduced to degenerate Lah-Bell polynomials if k 1 We derive their explicit expression, Dobinski-like formulas, recurrence relation, and various algebraic identities including Lah numbers, the degenerate Stirling numbers of the first and second kind, the degenerate poly-. E degenerate poly-Bernoulli polynomials are given by (see [4]). When x 0, β(nk,λ) β(nk,λ)(0) are called the degenerate poly-Bernoulli numbers. When x 0, G(nk,λ) G(nk,λ)(0) are called the degenerate poly-Genocchi numbers. For k ∈ Z, we introduced the degenerate polyBell polynomials bel(nk,λ)(x) given by (see [4]). El(nk,λ) bel(nk,λ)(1) are called the degenerate poly-Bell numbers. When λ ⟶ 0, bel(nk)(x) are called the poly-Bell polynomials

Degenerate Poly-Lah-Bell Polynomials and Numbers

Conclusion

Disclosure