Polynomial solutions of linear differential operators and Bochner's theorem

Consider linear differential operators of the form

\begin{equation*}\label{main.operator} \mathcal{L}_ru\stackrel{def}{=}\sum\limits_{j=0}^{r}Q_j(z)\dfrac{d^ju(z)}{dz^j}, \end{equation*}

where $\deg Q_j=n_j$, $j=0,1,\ldots,r$, and $Q_{0}(0)=0$.

In the talk, we discuss operators $\mathcal{L}_r$ having infinite sequences of polynomials eigenfunctions. We state necessary and sufficient conditions for such operators to have a complete sequences of eigenpolynomials and describe cases when those conditions fail. In particular, we found all the operators $\mathcal{L}_2$ with infinite sequences of polynomial eigenfunctions, including the cases missed by S. Bochner [1], and give examples of eigenpolynomial sequences for $\mathcal{L}_3$ and $\mathcal{L}_4$.

This is a joint work with Alexander Dyachenko.