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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

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    $\begingroup$ You might look at Bochner's classification of orthogonal polynomials that satisfy a second-order Sturm-Liouville equation. $\endgroup$ Dec 6, 2016 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? $\endgroup$ Dec 7, 2016 at 0:41
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    $\begingroup$ S. Bochner, Über Sturm-Liouvillesche Polynomsysteme. Math. Z. 29 (1929), 730-736. $\endgroup$ Dec 7, 2016 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, 2016 at 7:57

1 Answer 1

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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.

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  • $\begingroup$ There it is, in clear English. Thank you! $\endgroup$
    – Amir Sagiv
    Dec 12, 2016 at 7:57

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