Abstract
Umbral calculus is one of the important methods for obtaining the symmetric identities for the degenerate version of special numbers and polynomials. Recently, Kim–Kim (J. Math. Anal. Appl. 493(1):124521, 2021) introduced the λ-Sheffer sequence and the degenerate Sheffer sequence. They defined the λ-linear functionals and λ-differential operators, respectively, instead of the linear functionals and the differential operators of umbral calculus established by Rota. In this paper, the author gives various interesting identities related to the degenerate Lah–Bell polynomials and special polynomials and numbers by using degenerate Sheffer sequences, and at the same time derives the inversion formulas of these identities.
Highlights
1 Introduction It is important to note that many academics in the field of mathematics have been researching various degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects and in applications to differential equations, identities of symmetry and probability theory [9, 12, 14, 16,17,18,19,20,21,22,23], beginning with Carlitz’s degenerate Bernoulli polynomials and the degenerate Euler polynomials [2]
I focus on finding the noble identities of degenerate Lah–Bell polynomials in terms of quite a few well-known special polynomials
Kim–Kim introduced the degenerate Bell polynomial given by exλ eλ(t) – 1
Summary
It is important to note that many academics in the field of mathematics have been researching various degenerate versions of special polynomials and numbers in some arithmetic and combinatorial aspects and in applications to differential equations, identities of symmetry and probability theory [9, 12, 14, 16,17,18,19,20,21,22,23], beginning with Carlitz’s degenerate Bernoulli polynomials and the degenerate Euler polynomials [2].umbral calculus, established by Rota in the 1970s, was based on modern concepts such as linear functionals, linear operators, and adjoints [28]. (2020) 2020:687 and numbers arising from the degenerate Sheffer sequence. The author derives the inversion formulas of the identities obtained in this paper. The degenerate Bernoulli polynomials and degenerate Euler polynomials of order r, respectively, are given by the generating functions t eλ(t) – 1 r exλ(t) =
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