# Sturm Liouville problems for non-classical orthogonal polynomials

It is known that for the classical orthogonal-polynomials there exist a set of Sturm Liouville problems. E.g. , the Hermite polynomial of order $$n$$ is a solution of $$y''(x) -xy'(x)+ny(x)=0 \, .$$

My questions:

1. Given a finite measure $$\mu$$ on $$\Omega \subseteq \mathbb{R}$$, under what conditions does the respective set of orthogonal polynomials are a solution of respectve Sturm-Liouville problems?
2. In these cases, can we go back from $$\mu$$ to the differential equations, rather then from $$\mu$$ to the polynomials?

Inasmuch as I collected from Szego's book on Orthogonal Polynomials and some talks around campus, the answer to (1) is only the classical polynomials, and to (2) is no. But I didn't find any evidence of it.

EDIT - English Reference request: I was referred in the comments to Bochner's "Über Sturm-Liouvillesche Polynomsysteme" for a complete answer. Can anyone refer me to a translation or a book-chapter that gives its own version, in English?

Thanks

• You might look at Bochner's classification of orthogonal polynomials that satisfy a second-order Sturm-Liouville equation. – Robert Israel Dec 6 '16 at 23:26
• @Robert Israel: I think you are right, and there is a result saying that only classical ones afisfy a SL-problem. But what is a reference? – Alexandre Eremenko Dec 7 '16 at 0:41
• S. Bochner, Über Sturm-Liouvillesche Polynomsysteme. Math. Z. 29 (1929), 730-736. – Robert Israel Dec 7 '16 at 5:49
• @RobertIsrael Is there an English reference for that? A short google search didn't yield one. – Amir Sagiv Dec 7 '16 at 7:57