Characterization of symmetrical $H_q$-Laguerre-Hahn orthogonal polynomials of class zero

Abstract

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 $H_{q}$ be the $q$-derivative operator. We give the definition of the class $s$ of such form and the characterization of its corresponding orthogonal polynomials sequence $\{P_n\}_{n\geq0}$ by the structure relation. As a consequence, we establish the system fullfilled by the coefficients of the structure relation, those of the polynomials $\Phi, \Psi, B$ and the recurrence coefficients $\beta_{n}, \gamma_{n+1}, \, n \geq 0$ of $\{P_n\}_{n\geq0}$ for the class zero. In addition,  we carry out the complete description of the symmetrical $H_{q}$-Laguerre-Hahn forms of class $s=0.$ The limiting cases are also recovered.