Using D-operators to construct orthogonal polynomials satisfying higher order q-difference equations

Let $(p_n)_n$ be either the $q$-Meixner or the $q$-Laguerre polynomials. We form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+\beta_np_{n-1}$, $\beta_n\in \RR$. Using the concept of $\D$-operator, we generate sequences $(\beta_n)_n$ for which the polynomials $(q_n)_n$ are orthogonal with respect to a measure and common eigenfunctions of a higher order $q$-difference operator.