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

DOI:
https://doi.org/10.1090/S0002-9947-06-03901-8

Published electronically:
January 25, 2006

MathSciNet review:
2204051

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Abstract: An asymptotic expansion is derived for the Jacobi polynomials with varying parameters and , where and are constants. Our expansion is uniformly valid in the upper half-plane . A corresponding expansion is also given for the lower half-plane . 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 , 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 , and tend to as .

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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:
https://doi.org/10.1090/S0002-9947-06-03901-8

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.