Cross Recurrence Relations for \(r\)-Lah Numbers

Abstract

We give a new combinatorial interpretation of Lah and \(r\)-Lah numbers.
We establish two cross recurrence relations: the first one, which uses
an algebraic approach, is a recurrence relation of order two with
rational coefficients; the second one uses a combinatorial proof and
is a recurrence relation with integer coefficients. We also express
\(r\)-Lah numbers in terms of Lah numbers. Finally, we give identities
related to rising and falling factorial powers.