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

The n th 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.