On the classification of differential equations having orthogonal polynomial solutions — II

SummarySuppose {Φm(x)}is a given orthogonal polynomial sequence satisfying the differential equation(*) $$\sum\limits_{i - 1}^r {a_i (x)y^{(i)} (x) = \lambda y(x)} $$ . How does one construct an orthogonalizing weight distribution w(x) for {Φm(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 {Φm(x)} must also satisfy a second order differential equation of the form $$A(x,m)\Phi ''_m (x) + B(x,m)\Phi '_m (x) + C(x,m)\Phi _m (x) = 0$$ .