Structure relations of symmetric q-Dunkl classical orthogonal q-polynomials

期刊论文
发表日期：

2025年11月19日

查看全文

Y. Habbachi,
B. Bouras,
F. Marcellán

摘要

In this paper, we present two characterizations of symmetric q-Dunkl classical orthogonal q-polynomials. The first one is known in the literature as a first structure relation. Indeed, we can write the q-Dunkl derivative of a polynomial of degree n of a symmetric q-Dunkl polynomial sequence, multiplied by a polynomial of degree at most 2, as a finite linear combination of this sequence if and only if we are dealing with symmetric q-Dunkl classical orthogonal polynomials. The second one is known in the literature as a second structure relation. In this case, we can write a polynomial of degree n of a symmetric q-Dunkl sequence as a finite linear combination of the q-Dunkl derivative of this sequence if and only if we are dealing with symmetric q-Dunkl classical orthogonal polynomials. Finally, a connection between the Pearson equation associated with the q-Dunkl operator and the $H_{q}$ $H_{q}$ operator for symmetric q-Dunkl classical linear functionals is stated in order to prove that symmetric q-Dunkl classical orthogonal polynomial sequences are $H_{q}$ $H_{q}$ -semiclassical sequences of class $s=1.$ $s = 1 .$