On the classification of differential equations having orthogonal polynomial solutions (Q1061911)

Suppose \(\phi_ m\) is a polynomial of degree m that satisfies the differential equation (*) \(\sum^{2n}_{i=1}b_ i(x)y^{(i)}(x)=\lambda_ my(x)\) where n is some fixed integer \(\geq 1\). Then, under certain conditions, there exists an orthogonalizing weight distribution for a sequence \((\phi_ m)\) that simultaneously satisfies n distributional differential equations of orders 1,3,5,...(2n-1). In particular, this weight \(\Lambda\) must satisfy \(nb_{2n}\Lambda '+(nb_{2n}-b_{2n-1})\Lambda =0\) in the distributional sense. As a corollary to this result, the author gets part of \textit{H. L. Krall}'s classification theorem [Duke Math. J. 4, 705-718 (1938; Zbl 0020.02002)] which gives necessary and sufficient conditions on the existence of an orthogonal polynomial sequence of solutions to (*) in terms of the moments and the coefficients of \(b_ i\). The author illustrates the theory with all of the known orthogonal polynomial sequences to (*). In particular, new light is shed upon the problem of finding a real weight distribution for the Bessel polynomials.