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 | References | Similar Articles | Additional Information

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 .

**1.**Christof Bosbach and Wolfgang Gawronski,*Strong asymptotics for Jacobi polnomials with varying weights*, Methods Appl. Anal.**6**(1999), no. 1, 39–54. Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part I. MR**1803880**, 10.4310/MAA.1999.v6.n1.a3**2.**Li-Chen Chen and Mourad E. H. Ismail,*On asymptotics of Jacobi polynomials*, SIAM J. Math. Anal.**22**(1991), no. 5, 1442–1449. MR**1112518**, 10.1137/0522092**3.**P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, and X. Zhou,*Strong asymptotics of orthogonal polynomials with respect to exponential weights*, Comm. Pure Appl. Math.**52**(1999), no. 12, 1491–1552. MR**1711036**, 10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.3.CO;2-R**4.**P. Deift, S. Venakides, and X. Zhou,*New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems*, Internat. Math. Res. Notices**6**(1997), 286–299. MR**1440305**, 10.1155/S1073792897000214**5.**P. Deift and X. Zhou,*A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation*, Ann. of Math. (2)**137**(1993), no. 2, 295–368. MR**1207209**, 10.2307/2946540**6.**H. Dette and W. J. Studden,*Some new asymptotic properties for the zeros of Jacobi, Laguerre, and Hermite polynomials*, Constr. Approx.**11**(1995), no. 2, 227–238. MR**1342385**, 10.1007/BF01203416**7.**A. S. Fokas, A. R. It\cydots, and A. V. Kitaev,*The isomonodromy approach to matrix models in 2D quantum gravity*, Comm. Math. Phys.**147**(1992), no. 2, 395–430. MR**1174420****8.**C. L. Frenzen and R. Wong,*A uniform asymptotic expansion of the Jacobi polynomials with error bounds*, Canad. J. Math.**37**(1985), no. 5, 979–1007. MR**806651**, 10.4153/CJM-1985-053-5**9.**A. B. J. Kuijlaars, A. Martinez-Finkelshtein and R. Orive,*Orthogonality of Jacobi polynomials with general parameters*, Electr. Trans. Numer. Anal., to appear.**10.**Arno B. J. Kuijlaars and Kenneth T.-R. McLaughlin,*Riemann-Hilbert analysis for Laguerre polynomials with large negative parameter*, Comput. Methods Funct. Theory**1**(2001), no. 1, 205–233. MR**1931612**, 10.1007/BF03320986**11.**A. Martínez-Finkelshtein, P. Martínez-González, and R. Orive,*Zeros of Jacobi polynomials with varying non-classical parameters*, Special functions (Hong Kong, 1999) World Sci. Publ., River Edge, NJ, 2000, pp. 98–113. MR**1805976****12.**F. W. J. Olver,*Asymptotics and special functions*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR**0435697**

Frank W. J. Olver,*Asymptotics and special functions*, AKP Classics, A K Peters, Ltd., Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)]. MR**1429619****13.**Gábor Szegő,*Orthogonal polynomials*, 4th ed., American Mathematical Society, Providence, R.I., 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII. MR**0372517****14.**Nico M. Temme,*Large parameter cases of the Gauss hypergeometric function*, Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), 2003, pp. 441–462. MR**1985714**, 10.1016/S0377-0427(02)00627-1**15.**R. Wong,*Asymptotic approximations of integrals*, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989. MR**1016818****16.**R. Wong and Yu-Qiu Zhao,*Uniform asymptotic expansion of the Jacobi polynomials in a complex domain*, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.**460**(2004), no. 2049, 2569–2586. MR**2080215**, 10.1098/rspa.2004.1296

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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.