The (r1,...,rp)-Bell polynomials

TL;DR

This paper explores the properties of generalized Bell polynomials and Stirling numbers, establishing their log-concavity and providing new generating functions and recurrence relations.

Contribution

It introduces the log-concavity of $(r_{1},...,r_{p})$-Stirling numbers and derives new generating functions and recurrences for the generalized Bell polynomials.

Findings

$(r_{1},...,r_{p})$-Stirling numbers are log-concave

New generating functions for generalized Bell polynomials

Generalized recurrence relations derived

Abstract

In a previous paper, Mihoubi et al. introduced the $(r_{1},...,r_{p}) $-Stirling numbers and the $(r_{1},...,r_{p}) $-Bell polynomials and gave some of their combinatorial and algebraic properties. These numbers and polynomials generalize, respectively, the $r$-Stirling numbers of the second kind introduced by Broder and the $r$-Bell polynomials introduced by Mez\H{o}. In this paper, we prove that the $(r_{1},...,r_{p}) $-Stirling numbers of the second kind are log-concave. We also give generating functions and generalized recurrences related to the $(r_{1},...,r_{p}) $-Bell polynomials.