Description of the symmetric $$H_q$$ -Laguerre–Hahn orthogonal q-polynomials of class one

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