Some Identities on λ-Analogues of Lah Numbers and Lah-Bell Polynomials

In recent years, some applications of Lah numbers were discovered in the real world problem of telecommunications and optics. The aim of this paper is to study the λ-analogues of Lah numbers and Lah-Bell polynomials which are λ-analogues of the Lah numbers and and Lah-Bell polynomials. Here we note that λ-analogues appear when we replace the falling factorials by the generalized falling factorials in the defining equations. By using generating function method, we study some properties, explicit expressions, generating functions and Dobinski-like formulas for those numbers and polynomials. We also treat the more general λ-analogues of r-Lah numbers and r-extended λ-Lah-Bell polynomials. In addition, we show that the expectations of two random variables, both associated with the Poisson random variable with parameter α , are equal to the λ-analogue of the Lah-Bell polynomial evaluated at α for one and the r-extended λ-Lah-Bell polynomial evaluated at α for the other.