New q-Laguerre polynomials having factorized permutation interpretations

Abstract

In this paper, we generalize the Laguerre polynomials in terms of q-analogue for Riordan matrices. To be more specific, for a is an element of N-0, we introduce new q-Laguerre polynomials L-n((alpha))(x; q) by defining the Eulerian generating function for L-n((alpha))(x; q) as Pi(alpha)(j=0) 1/1+q(z)(j+1)) e(q) [xz/1+z]. Interestingly, it turns out that L-n((alpha)) (x; q) have combinatorial descriptions in the aspect of the inversions of factorized permutations and q-rook numbers. We locate their zeros and develop their algebraic properties as well. (C) 2018 Elsevier Inc. All rights reserved.