Some identities of fully degenerate Dowling and fully degenerate Bell polynomials arising from lambda-umbral calculus

TL;DR

This paper introduces and explores properties of fully degenerate Bell and Dowling polynomials within the lambda-umbral calculus framework, extending classical polynomial families and their combinatorial interpretations.

Contribution

It presents new fully degenerate versions of Bell and Dowling polynomials and investigates their identities using lambda-umbral calculus, expanding the theoretical understanding of these polynomials.

Findings

Derived identities relating fully degenerate Bell and Dowling polynomials.

Established connections between degenerate polynomials and classical combinatorial numbers.

Extended the Whitney numbers of the second kind through degenerate polynomial versions.

Abstract

Recently, Kim-Kim introduced the lambda-umbral calculus, in which the lambda-Sheffer sequences occupy the central position. In this paper, we introduce the fully degenerate Bell and the fully degenerate Dowling polynomials, and investigate some properties and identities relating to those polynomials with the help oflambda-umbral calculus. Here we note that the fully degenerate Bell poynomials and the fully degenerate Dowling polynomials are respectively degenerate versions of the Bell polynomials and the Dowling polynomials, of which the latters are the natural extension of the Whitney numbers of the second kind.