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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Uniform asymptotics for Jacobi polynomials with varying large negative parameters-- a Riemann-Hilbert approach


Authors: R. Wong and Wenjun Zhang
Journal: Trans. Amer. Math. Soc. 358 (2006), 2663-2694
MSC (2000): Primary 41A60, 33C45
Published electronically: January 25, 2006
MathSciNet review: 2204051
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Abstract: An asymptotic expansion is derived for the Jacobi polynomials $ P_{n}^{(\alpha_{n},\beta_{n})}(z)$ with varying parameters $ \alpha_{n}=-nA+a$ and $ \beta_n=-nB+b$, where $ A>1, B>1$ and $ a,b$ are constants. Our expansion is uniformly valid in the upper half-plane $ \overline{\mathbb{C}}^+=\{z:{Im}\; z \geq 0\}$. A corresponding expansion is also given for the lower half-plane $ \overline{\mathbb{C}}^-=\{z:{Im}\; z \leq 0\}$. Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993). The two asymptotic expansions hold, in particular, in regions containing the curve $ L$, which is the support of the equilibrium measure associated with these polynomials. Furthermore, it is shown that the zeros of these polynomials all lie on one side of $ L$, and tend to $ L$ as $ n \to \infty$.


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Additional Information

R. Wong
Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Wenjun Zhang
Affiliation: Department of Mathematics, Normal College, Shenzhen University, Shenzhen, Guang-dong, People’s Republic of China, 518060
Email: zwj@szu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03901-8
PII: S 0002-9947(06)03901-8
Keywords: Jacobi polynomials, Riemann-Hilbert problem, uniform asymptotics, zero distribution, equilibrium measure
Received by editor(s): August 3, 2004
Published electronically: January 25, 2006
Additional Notes: The work of the first author was partially supported by the Research Grant Council of Hong Kong under project 9040980, and the work of the second author was partially supported by the National Natural Science Foundation of China
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.