Description of the symmetric \(H_q\)-Laguerre–Hahn orthogonal q-polynomials of class one

We study the \(H_{q}\) $H_q$ -Laguerre–Hahn forms u, that is to say those satisfying a q-quadratic q-difference equation with polynomial coefficients ( \(\Phi , \Psi , B\) $\mathrm{\Phi },\mathrm{\Psi },B$ ): \( H_{q}(\Phi (x)u) +\Psi (x) u+B(x) \, \big (x^{-1}u(h_{q}u)\big )=0,\) $H_q(\mathrm{\Phi }(x)u)+\mathrm{\Psi }(x)u+B(x)(x^{-1}u(h_{q}u))=0,$ where \(h_q u\) $h_{q}u$ is the form defined by \(\langle h_{q} u,f\rangle =\langle u, f(qx)\rangle \) $⟨h_{q}u,f⟩=⟨u,f(qx)⟩$ for all polynomials f and \(H_{q}\) $H_q$ is the q-derivative operator. We give the definition of the class s of such a form and the characterization of its corresponding orthogonal q-polynomials sequence \(\{P_n\}_{n\ge 0}\) ${\{P_n\}}_{n\ge 0}$ by the structure relation. As a consequence, we establish the system fulfilled by the coefficients of the structure relation, those of the polynomials \(\Phi , \Psi , B\) $\mathrm{\Phi },\mathrm{\Psi },B$ and the recurrence coefficient \(\gamma _{n+1}, \, n \ge 0\) ${\gamma }_{n+1},n\ge 0$ , of \(\{P_n\}_{n\ge 0}\) ${\{P_n\}}_{n\ge 0}$ for the class one in the symmetric case. In addition, we present the complete description of the symmetric \(H_{q}\) $H_q$ -Laguerre–Hahn forms of class \(s=1.\) $s=1.$ The limiting cases are also covered.