Differential equations having orthogonal polynomial solutions

Necessary and sufficient conditions for an orthogonal polynomial system (OPS) to satisfy a differential equation with polynomial coefficients of the form (∗) LN[y] = ∑i=1Nli(x)y(i)(x) = λny(x) were found by H.L. Krall. Here, we find new necessary conditions for the equation (∗) to have an OPS of solutions as well as some other interesting applications. In particular, we obtain necessary and sufficient conditions for a distribution w(x) to be an orthogonalizing weight for such an OPS and investigate the structure of w(x). We also show that if the equation (∗) has an OPS of solutions, which is orthogonal relative to a distribution w(x), then the differential operator LN[·] in (∗) must be symmetrizable under certain conditions on w(x).