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?


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    $\begingroup$ You might look at Bochner's classification of orthogonal polynomials that satisfy a second-order Sturm-Liouville equation. $\endgroup$ – Robert Israel Dec 6 '16 at 23:26
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    $\begingroup$ @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? $\endgroup$ – Alexandre Eremenko Dec 7 '16 at 0:41
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    $\begingroup$ S. Bochner, Über Sturm-Liouvillesche Polynomsysteme. Math. Z. 29 (1929), 730-736. $\endgroup$ – Robert Israel Dec 7 '16 at 5:49
  • $\begingroup$ @RobertIsrael Is there an English reference for that? A short google search didn't yield one. $\endgroup$ – Amir Sagiv Dec 7 '16 at 7:57

A reference in english for Bochner's theorem is section 20.1, p.508, of the book by Mourad E.H.Ismail,

Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, 98. Cambridge University Press, Cambridge, 2009.

  • $\begingroup$ There it is, in clear English. Thank you! $\endgroup$ – Amir Sagiv Dec 12 '16 at 7:57

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