Complete and incomplete Bell polynomials associated with Lah-Bell numbers and polynomials

TL;DR

This paper introduces incomplete and complete r-extended Lah-Bell polynomials, generalizing Lah-Bell numbers, and explores their properties and identities to provide new combinatorial expressions.

Contribution

It presents the first multivariate versions of r-extended Lah-Bell numbers and polynomials, expanding their theoretical framework and deriving new identities.

Findings

Derived finite sum expressions for r-Lah numbers

Established properties and identities of the new polynomials

Connected polynomials to combinatorial set partitions

Abstract

The nth r-extended Lah-Bell number is defined as the number of ways a set with $n+r$ elements can be partitioned into ordered blocks such that r distinguished elements have to be in distinct ordered blocks. The aim of this paper is to introduce incomplete r-extended Lah-Bell polynomials and complete $r$-extended Lah-Bell polynomials respectively as multivariate versions of $r$-Lah numbers and the r-extended Lah-Bell numbers and to investigate some properties and identities for these polynomials. From these investigations, we obtain some expressions for the r-Lah numbers and the r-extended Lah-Bell numbers as finite sums.