Differential equations having orthogonal polynomial solutions (Q1360153)

The authors study differential operators of the form \[ L_N[y]:=\sum_{i=1}^n\ell_i(x)y^{(i)}(x), \] with polynomial coefficients \(\ell_i(x)\). Necessary and sufficient conditions that the differential operator has as eigenfunctions an orthogonal polynomial sequence (i.e. \(\{P_n(x)\}\) with \(\text{deg } P_n=n\) and \(L_N[P_n(x)]=\lambda_n P_n(x)\), \(n\geq 0\)) were already given by \textit{S. Bochner} for \(N=2\) [Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29, 730-736 (1929; JFM 55.0260.01] and for general \(N\) by \textit{H. L. Krall} [Certain differential equations for Chebychev polynomials, Duke Math. J. 4, 705-718 (1938; Zbl 0020.02002)]. H. L. Krall also gave a complete classification in the case of \(N=4\). For \(N>4\) the complete classification of the appropriate differential operators still remains open, except for several `isolated' cases (see the references in the paper). The authors now obtain new (and simpler) necessary conditions for the existence of an orthogonal polynomial sequence and moreover give necessary and sufficient conditions on a distribution \(w(x)\) to be an orthogonalizing weight. Finally a connection between the existence of the polynomials and the symmetrizability of the underlying differential operator is looked into.