DOI: 10.2478/s12175-013-0116-3 Math. Slovaca 63 (2013), No. 3, 531–544 THE LAGUERRE POLYNOMIALS IN SEVERAL VARIABLES Rab˙ ia Aktas ¸* — Esra Erkus ¸-Duman** (Communicated by J´ an Bors´ ık ) ABSTRACT. In this paper, we give some relations between multivariable La- guerre polynomials and other well-known multivariable polynomials. We get var- ious families of multilinear and multilateral generating functions for these poly- nomials. Some special cases are also presented. c 2013 Mathematical Institute Slovak Academy of Sciences 1. Introduction As usual, let r F s denote the generalized hypergeometric function with r nu- merator and s denominator parameters. Then, the classical Laguerre polynomi- als L (α) n (x) of degree n are given by the Rodrigues formula [8] L (α) n (x)= x -α e x n! D n x ( e -x x n+α ) with D n x := d n dx n , or, equivalently, by L (α) n (x)= (1 + α) n n! 1 F 1 (-n; α + 1; x) . (1.1) It is well-known that these polynomials are orthogonal over the interval (0, ∞) with respect to the weight function ω (x)=e -x x α for Re(α) > -1. Actually, we have ∞  0 e -x x α L (α) n (x) L (α) m (x)dx = Γ(α + n + 1) n! δ m,n , 2010 Mathematics Subject Classification: Primary 33C45. K e y w o r d s: generating function, hypergeometric function, multivariable Laguerre polynomi- als, multilinear and multilateral generating functions.