Difference equations for the co-recursive \(r\)th associated orthogonal polynomials of the \(D _{q}\)-Laguerre-Hahn class.
A family of orthogonal polynomials \((P_n)_n\) is said to be of the \(D_q\)-Laguerre-Hahn class if the corresponding Stieltjes function satisfies a polynomial Riccati \(q\)-difference equation \(Ay\sigma_qy+B\sigma_qy+Cy+D=0\), where \(A\), \(B\), \(C\) and \(D\) are polynomials and where \(\sigma_q\) is the \(q\)-difference operator given by \((\sigma_q y)(x)=y(qx)\). The set of Riccati \(q\)-difference equations is invariant w.r.t. any homographic transform on \(y\). So, all co-recursive \(r\)th associated orthogonal polynomials \((P_n^{\{r,\mu\}})_n\) are of the \(D_q\)-Laguerre-Hahn class if \((P_n)_n\) is supposed to be in this class. Using the functional relations between \((P_n)_n\) and \((P_n^{\{r,\mu\}})_n\), the authors give a fourth-order linear q-difference equation satisfied by \((P_n^{r,\mu})_n\). Some applications are also presented.
scientific article; zbMATH DE number 1915235