$d$-orthogonality of Little $q$-Laguerre type polynomials

Abstract

In this paper, we solve a characterization problem in the context of the $d$-orthogonality. That allows us, on one hand, to provide a $q$-analog for the $d$-orthogonal polynomials of Laguerre type introduced by the first author and Douak, and on the other hand, to derive new $L_q$-classical $d$-orthogonal polynomials generalizing the Little $q$-Laguerre polynomials. Various properties of the resulting basic hypergeometric polynomials are singled out. For $d=1$, we obtain a characterization theorem involving, as far as we know, new $L_q$-classical orthogonal polynomials, for which we give the recurrence relation and the difference equation.