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

DOI:
https://doi.org/10.1090/S0002-9939-2010-10601-9

Published electronically:
October 18, 2010

MathSciNet review:
2763762

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

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.