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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

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New integral identities for orthogonal polynomials on the real line
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by D. S. Lubinsky PDF
Proc. Amer. Math. Soc. 139 (2011), 1743-1750 Request permission

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 $\leq 2n-2$, \begin{equation*} \int _{-\infty }^{\infty }\frac {P(t)}{p_{n}^{2}\left ( t\right ) } h\left ( \frac {p_{n-1}}{p_{n}} \left ( t\right ) \right ) dt=\frac {\gamma _{n-1}}{\gamma _{n}} \left ( \int _{-\infty }^{\infty }h\left ( t\right ) dt\right ) \left ( \int P\left ( t\right ) \text { }d\mu \left ( t\right ) \right ) . \end{equation*} 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
  • MR Author ID: 116460
  • ORCID: 0000-0002-0473-4242
  • Email: lubinsky@math.gatech.edu
  • 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
  • © Copyright 2010 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 1743-1750
  • MSC (2010): Primary 42C05
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10601-9
  • MathSciNet review: 2763762