In this paper, we give two characterization theorems to introduce new examples of basic hypergeometric d -orthogonal polynomials which, on the one hand, generalize the known q-Meixner, big q-Laguerre, little q-Laguerre and q-Laguerre polynomials and, on the other hand, can be viewed as q-analogs of d -orthogonal polynomials of Meixner and Laguerre type. For the resulting polynomials, we state some properties.
In this paper, we treat three questions related to the d-orthogonality of the Humbert polynomials. The first one consists to determinate the explicit expression of the d-dimensional functional vector for which the d-orthogonality holds. The second one is the investigation of the components of Humbert polynomial sequence. That allows us to introduce, as far as we know, new d-orthogonal polynomials generalizing the classical Jacobi ones. The third one consists to solve a characterization problem related to a generalized hypergeometric representation of the Humbert polynomials.
In this paper, we characterize the d-orthogonal polynomial sets given by their explicit expressions in a specific basis. As application, we consider the generalized hypergeometric case to characterize d-orthogonal polynomial sets of Laguerre type, Meixner type, Meixner-Pollaczek type, Krawtchouk type, continuous dual Hahn type, and dual Hahn type. For d = 1, we obtain a unification of some characterization theorems in the orthogonal polynomials theory.
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