**Ali Zaghouani**

## Some Basic $d$-Orthogonal Polynomial
Sets

**Abstract:**

The purpose of this paper is to study the class of polynomial sets which are at
the same time $d$-orthogonal and $q$-Appell. By a linear change of variable, the
resulting set reduces to $q$-Al-Salam-Carlitz polynomials, for $d=1$. Various
properties of the obtained polynomials are singled out: a generating function, a
recurrence relation of order $d+1$. We also explicitly express a $d$-dimensional
functional for which the d-orthogonality holds.

**Keywords:**

Orthogonal polynomials, $d$-orthogonal polynomials, $d$-dimensional functional,
$q$-difference operator.

**MSC 2000:** 42C05, 33D45