\(q\)-coherent pairs and \(q\)-orthogonal polynomials
The concept of \(q\)-coherent pair of linear functionals is introduced and it is proved that if \((u_0,u_1)\) is a coherent pair, then both \(u_0\) and \(u_1\) are \(q\)-semiclassical linear functionals. It is also proved that if \((u_0,u_1)\) is a \(q\)-coherent pair of linear functionals, then at least one of them must be a \(q\)-classical linear functional. The classification of all \(q\)-coherent pairs of positive-definite linear functionals \((u_0,u_1)\) when \(u_0\) or \(u_1\) is either the little \(q\)-Jacobi linear functional or the little \(q\)-Laguerre linear functional is given. Finally, by using limit relations, the classification of all coherent pairs of positive definite linear functionals is recovered.
scientific article; zbMATH DE number 1861069
scientific article