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

Abstract

Abstract The nth r-extended Lah–Bell number is defined as the number of ways a set with n + r $n+r$ 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.

Keywords

• Incomplete r-extended Lah–Bell polynomial
• Complete r-extended Lah–Bell polynomial