Spivey's type recurrence relation for degenerate Bell polynomials

TL;DR

This paper derives a generalized recurrence relation for degenerate Bell polynomials using operator methods, extending Spivey's relation for Bell numbers, and also applies these techniques to degenerate r-Bell polynomials.

Contribution

It introduces a new operator-based approach to derive recurrence relations for degenerate Bell and r-Bell polynomials, generalizing existing formulas.

Findings

Derived a recurrence relation for degenerate Bell polynomials.

Extended Spivey's recurrence to degenerate r-Bell polynomials.

Provided operator-based proofs for the relations.

Abstract

The aim of this paper is to derive a recurrence relation for the degenerate Bell polynomials by using the operators X and D satisfying the commutation relation DX-XD=1. Here X is the `multiplication by x' operator and D=d/dx. This recurrence relation is a generalization of Spivey's recurrence relation for the Bell numbers. We also obtain a recurrence relation for the degenerate r--Bell polynomials by using the same operators.