A Generalized Recurrence for fully degenerate Bell polynomials

TL;DR

This paper introduces a new family of fully degenerate Bell polynomials and their two-variable versions, deriving natural recurrence relations and operator expressions, extending to r-counterparts with Dobinski-like formulas.

Contribution

It presents a novel family of fully degenerate Bell polynomials with natural recurrence relations and operator formulas, extending to r-counterparts.

Findings

Derived natural Spivey-type recurrence relations.

Extended results to r-counterparts with Dobinski-like formulas.

Provided operator expressions for all new polynomials.

Abstract

This paper addresses the unnatural appearance of the two-variable degenerate Fubini polynomials in a recently derived Spivey-type recurrence relation for the fully degenerate Bell polynomials. To solve this, we introduce a new family of polynomial which we also call the fully degenerate Bell polynomials, along with their two-variable counterparts. Our main contribution is the derivation of natural Spivey-type recurrence relations using operator methods. We extend these results to the r-counterparts, the fully degenerate r-Bell polynomials providing Dobinski-like, finite sum, operator expressions, and Spivey-type recurrence relations for all the new polynomials.