  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.
