The Bochner-Krall problem: Some new perspectives (Q2760154)

The Bochner-Krall problem is to find all families of orthogonal polynomials that are eigenfunctions of a differential operator [see \textit{S. Bochner}, Math. Z. 29, 730-736 (1929; JFM 55.0260.01)].NEWLINENEWLINENEWLINEThe article represents a survey of some new results in the sphere of the extended Bochner-Krall problem, in particular, on the subject of spectral matrix (see [\textit{F. A. Grünbaum} and \textit{L. Haine}, Methods Appl. Anal. 6, II. Ser. 209-224 (1999; Zbl 0956.33007)], which have been received with the help of a new method developed by the author. It is shown the connection between classical orthogonal polynomials and the Duistermaat-Grünbaum problem, and the Darboux factorization method. NEWLINENEWLINENEWLINEThe author offers some solutions of the extended Bochner-Krall problem besides the Hermite, the Laguerre, the Jacobi and the Bessel classical polynomials. It is expressed a conjecture that there exist no other solutions of the subject problem.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00053].