The $r$-Bell numbers

TL;DR

This paper provides a comprehensive and systematic study of generalized Bell numbers, exploring their combinatorial, algebraic, and analytic properties, and introduces the related $r$-Bell polynomials, filling a gap in the literature.

Contribution

It offers the first systematic treatise on generalized Bell numbers, detailing their properties and introducing $r$-Bell polynomials, with many results appearing to be new.

Findings

Generalized Bell numbers have rich combinatorial and algebraic properties.

Introduction of $r$-Bell polynomials expands the theory.

Most results are novel and unify various existing observations.

Abstract

The notion of generalized Bell numbers has appeared in several works but there is no systematic treatise on this topic. In this paper we fill this gap. We discuss the most important combinatorial, algebraic and analytic properties of these numbers which generalize the similar properties of the Bell numbers. Most of these results seem to be new. It turns out that in a paper of Whitehead these numbers appeared in a very different context. In addition, we introduce the so-called $r$-Bell polynomials.