Spivey's Bell Number Formula Revisited

TL;DR

This paper revisits Spivey's Bell number formula by deriving an alternative form using $q$-Boson operators and extends the approach to $(q,r)$-Dowling polynomials, providing a broader generalization.

Contribution

It introduces a new derivation of Spivey's Bell number formula using $q$-Boson operators and generalizes it to $(q,r)$-Dowling polynomials.

Findings

Derived an alternative form of Spivey's Bell number formula

Extended the formula to $(q,r)$-Dowling polynomials

Produced a generalized version applicable to broader polynomial classes

Abstract

This paper introduces an alternative form of the derivation of Spivey's Bell number formula which involves the $q$-Boson operators $a$ and $a^{\dagger}$. Furthermore, a similar formula for the case of the $(q,r)$-Dowling polynomials is obtained, and is shown to produce a generalization of the latter.