On the classification of differential equations having orthogonal polynomial solutions. II (Q1101860)

Suppose \(\{\Phi_ m(x)\}\) is a given orthogonal polynomial sequence satisfying the differential equation \[ (*)\quad \sum^{r}_{i=1}a_ i(x)y^{(i)}(x) = \lambda y(x). \] How does one construct an orthogonalizing weight distribution w(x) for \(\{\Phi_ m(x)\}?\) We answer this question in this paper as well as show some interesting new applications. In particular, we apply a result of Atkinson and Everitt to show that \(\{\Phi_ m(x)\}\) must also satisfy a second order differential equation of the form \[ A(x,m)\Phi ''_ m(x)\quad +\quad B(x,m)\Phi '_ m(x)\quad +\quad C(x,m)\Phi_ m(x) = 0. \] We also show how the techniques developed lead to a new set of orthogonal polynomials and to some insight on symmetric orthogonal polynomials satisfying a self adjoint equation of the form (*). This article is a continuation of ibid. 4, 35-53 (1984; Zbl 0571.34003); we extend and improve upon these results.