Degenerate Lah–Bell polynomials arising from degenerate Sheffer sequences

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.

Central factorial numbers associated with sequences of polynomials

“…Then x n ∼ (1, t). By the definition in (3.5) and Theorem 3.1, we have (see earlier studies [2,10,24]) the following: [18,23,[25][26][27][28][29]). By Theorem 3.1, the generating function is given by…”

“…}{\left({e}^{\frac{t}{2}}-{e}^{-\frac{t}{2}}\right)}^k, $$ (see ()). Let $$$ \boldsymbol{P}=\left\{{(x)}_{n,\lambda}\right\} $$$ be the sequence of generalized falling factorials. Then $$$ {(x)}_{n,\lambda}\sim \left(1,f(t)=\frac{1}{\lambda}\left({e}^{\lambda t}-1\right)\right) $$$, with $$$ \overline{f}(t)=\frac{1}{\lambda}\log \left(1+\lambda t\right) $$$ (see earlier research [18, 23, 25–29]).By Theorem 3.1, the generating function is given by $$$$ \sum \limits_{n=k}^{\infty }{T}_2\left(n,k;\boldsymbol{P}\right)\frac{t^n}{n!}=\frac{1}{k! }\left({e}_{\lambda}^{\frac{1}{2}}(t)-{e}_{\lambda}^{-\frac{1}{2}}(t)\right)=\sum \limits_{n=k}^{\infty }{T}_{2,\lambda}\left(n,k\right)\frac{t^n}{n!$$…”

The \(\alpha\)-analogues of r-Whitney Numbers via Normal Ordering

“…Recently, some mathematicians have studied the counting sequences problems derived from Boson operators including the λ-analogues of r-Stirling numbers of the second kind, the analogue of r-Stirling numbers, the degenerate Stirling numbers, the degenerate r-Bell polynomials, the degenerate r-Whitney numbers and degenerate r-Dowling polynomials (see [8,[11][12][13][14][15][16][17][18][19]).…”

A Study on degenerate Whitney numbers of the first and second kinds of Dowling lattices

“…Here we remark that the study of degenerate versions of some special numbers and polynomials, which was initiated by Carlitz in [6,7], regained the interests of some mathematicians and yielded many interesting results. For some of these, one refers to [1,[11][12][13][14][15][16][17][18][19]22 and references therein]. Here we would like to mention only two things.…”

“…Let log λ (t) be the compositional inverse of e λ (t), called the degenerate logarithms. Then we have (13) log…”