New tools for the study of Bochner differential operators

A sequence $\{{\delta }_n^{(k)}\}$ associated to a Bochner differential operator is introduced as an effective tool to study this kind of operators. Some properties of this sequence are proven and used to deduce that a particular operator leads to solutions of a bispectral problem. In addition, the inverse problem is studied; that is, given a sequence $\{{\lambda }_n\}$ of complex numbers and a sequence $\{P_n\}$ of polynomials with complex coefficients, $\mathrm{°}{P}_n=n$, we find a necessary and sufficient condition for the existence of a Bochner differential operator that has those sequences as eigenvalues and eigenpolynomials, respectively. The mentioned condition also depends on $\{{\delta }_n^{(k)}\}$.