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

 

Asymptotics for Sobolev orthogonal polynomials with coherent pairs: The Jacobi case, type 1


Author: K. Pan
Journal: Proc. Amer. Math. Soc. 126 (1998), 2377-2388
MSC (1991): Primary 42C05
MathSciNet review: 1443401
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Abstract: Define $P_n(x)$ and $Q_n(x)$ as the $n$th monic orthogonal polynomials with respect to $d\mu$ and $d\nu$ respectively. The pair $\{d\mu,d\nu\}$ is called a coherent pair if there exist non-zero constants $D_n$ such that

\begin{displaymath}Q_n(x)=\frac{P_{n+1}^\prime(x)}{n+1}+D_n\frac{P_n^\prime(x)}{n},\qquad n\ge 1.\end{displaymath}

One can divide the coherent pairs into two cases: the Jacobi case and the Laguerre case. There are two types for each case: type 1 and 2. We investigate the asymptotic properties and zero distribution of orthogonal polynomials with respect to Sobolev inner product

\begin{displaymath}\langle f,g\rangle =\int _a^b f(x)g(x)d\mu(x)+\lambda \int _a^b f'(x)g'(x)d\nu(x)\end{displaymath}

for the coherent pair $\{d\mu,d\nu\}$: the Jacobi case, type 1.


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Additional Information

K. Pan
Affiliation: Department of Mathematics, Barry University, Miami Shores, Florida 33161
Email: pan@euclid.barry.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04300-7
PII: S 0002-9939(98)04300-7
Received by editor(s): July 24, 1996
Received by editor(s) in revised form: January 22, 1997
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society