New q-Laguerre polynomials having factorized permutation interpretations

In this paper, we generalize the Laguerre polynomials in terms of q-analogue for Riordan matrices. To be more specific, for α ∈ N 0 , we introduce new q-Laguerre polynomials L n ( α ) ( x ; q ) by defining the Eulerian generating function for L n ( α ) ( x ; q ) as ( ∏ j = 0 α 1 1 + q j + 1 z ) e q [ x z 1 + z ] . Interestingly, it turns out that L n ( α ) ( 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.