The quasi-orthogonality of the derivatives of semi-classical polynomials

The distributional equation of a semi-classical functional allows an efficient study of other characterizations and properties of the semi-classical OPS/functionals. In 6] an extensive survey of this approach has been presented. A particular case of semi-classical OPS/functionals are the classical ones. For the distributional equation of a classical functional a regularity condition holds. We have give a family counterexamples to show that the regularity condition does not hold in general for semi-classical functionals 9]. Here we investigate the consequences of the failure of the regularity condition in the quasi-orthogonality of the derivatives of the semi-classical OPS and, in general, the behaviour of the derivatives of order k. This study leads us to another condition that holds for the distributional equation of classical functional, the coprimality condition.