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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

New integral identities for orthogonal polynomials on the real line


Author: D. S. Lubinsky
Journal: Proc. Amer. Math. Soc. 139 (2011), 1743-1750
MSC (2010): Primary 42C05
Published electronically: October 18, 2010
MathSciNet review: 2763762
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Abstract: Let $ \mu $ be a positive measure on the real line, with associated orthogonal polynomials $ \left\{ p_{n}\right\} $ and leading coefficients $ \left\{ \gamma _{n}\right\} $. Let $ h\in L_{1}\left( \mathbb{R}\right) $ . We prove that for $ n\geq 1$ and all polynomials $ P$ of degree $ \leq 2n-2$,

$\displaystyle \int_{-\infty }^{\infty }\frac{P(t)}{p_{n}^{2}\left( t\right) } h... ... dt\right) \left( \int P\left( t\right) \text{ }d\mu \left( t\right) \right) . $

As a consequence, we establish weak convergence of the measures on the left-hand side.


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Additional Information

D. S. Lubinsky
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: lubinsky@math.gatech.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10601-9
PII: S 0002-9939(2010)10601-9
Keywords: Orthogonal polynomials on the real line, Geronimus type formula, Poisson integrals
Received by editor(s): March 23, 2010
Received by editor(s) in revised form: May 21, 2010
Published electronically: October 18, 2010
Additional Notes: This research was supported by NSF grant DMS1001182 and U.S.-Israel BSF grant 2008399
Communicated by: Walter Van Assche
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.