A new approach to Bell and poly-Bell numbers and polynomials

TL;DR

This paper introduces new classes of Bell and poly-Bell polynomials and numbers, derives their explicit formulas, recurrence relations, identities, and explores their degenerate versions, expanding the mathematical understanding of these special functions.

Contribution

It presents novel definitions, explicit expressions, and identities for Bell and poly-Bell polynomials and their degenerate forms, which were not previously studied.

Findings

Derived explicit formulas for Bell and poly-Bell polynomials.

Established recurrence relations and identities for these polynomials.

Introduced and analyzed degenerate versions of Bell and poly-Bell polynomials.

Abstract

The aim of this paper is to introduce Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. We also consider degenerate versions of those polynomials and numbers, namely degenerate Bell polynomials and numbers of the second kind and degenerate poly-Bell polynomials and numbers of the second kind, and deduce their similar results.