Some Identities on λ-Analogues of Lah Numbers and Lah-Bell Polynomials (European Journal of Pure and Applied Mathematics, 2024)
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.
Keywords: λ-analogues of Lah numbers; λ-analogues of Lah-Bell polynomials; λ- analogues of Laguerre polynomials; λ-analogues of r-numbers; r-extended λ-Lah-Bell polynomials