Characterization of orthogonal polynomials – a new proof of Bochner’s theorem. (English) Zbl 1402.33009
Agranovsky, Mark L. (ed.) et al., Complex analysis and dynamical systems VII. Proceedings of the 7th international conference on complex analysis and dynamical systems (CA&DS VII), Nahariya, Israel, May 10–15, 2015. Providence, RI: American Mathematical Society (AMS); Ramat Gan: Bar-Ilan University (ISBN 978-1-4704-2961-4/pbk; 978-1-4704-4256-9/ebook). Contemporary Mathematics 699. Israel Mathematical Conference Proceedings, 87-101 (2017).
Summary: In this paper we characterize classes of orthogonal polynomials, which brings us naturally to present a new proof of Bochner’s theorem. Our main concern is the classifications of the weight functions in the situation when a system of orthogonal polynomials satisfies the Sturm-Liouville equation. Moreover, we show that the Jacobi, Laguerre and Hermite systems of orthogonal polynomials are the only ones which satisfy the condition that both they and there derivatives are a system of orthogonal polynomials.
For the entire collection see [Zbl 1394.00019].
For the entire collection see [Zbl 1394.00019].
MSC:
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
| 34B24 | Sturm-Liouville theory |
| 47E05 | General theory of ordinary differential operators |
Keywords:
characterization; Sturm-Liouville equation; Bochner’s theorem; differential operator; orthogonal polynomialsReferences:
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