Some Generalizations of Spivey's Bell Number Formula

TL;DR

This paper introduces a generalized recurrence relation for the r-Whitney numbers of the second kind, extending Spivey's Bell number formula, and presents additional related identities within this framework.

Contribution

It derives a new generalized recurrence relation for r-Whitney numbers using operator methods, expanding the scope of Spivey's Bell number generalizations.

Findings

Derived a generalized recurrence relation for r-Whitney numbers

Established new identities extending Spivey's Bell number formula

Unified operator approach using X and D satisfying [D,X]=1

Abstract

In this paper, a generalized recurrence relation for the $r$-Whitney numbers of the second kind is derived using as framework the operators $X$ and $D$ satisfying the commutation relation $DX-XD=1$. This recurrence relation is shown to be a generalization of the well-known Spivey's Bell number. Moreover, several other identities generalizing Spivey's Bell number formula are obtained.