AN ALTERNATIVE RIEMANN-HILBERT APPROACH AND ITS APPLICATION TO THE GLOBAL ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS

Jie Yan; Huili Han; Min Ku; Xin Wei

doi:10.11948/20250266

2026 Volume 16 Issue 6

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Article navigation > Journal of Applied Analysis & Computation > 2026 > 16(6): 3279-3306

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Jie Yan, Huili Han, Min Ku, Xin Wei. AN ALTERNATIVE RIEMANN-HILBERT APPROACH AND ITS APPLICATION TO THE GLOBAL ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3279-3306. doi: 10.11948/20250266

Citation:

Jie Yan, Huili Han, Min Ku, Xin Wei. AN ALTERNATIVE RIEMANN-HILBERT APPROACH AND ITS APPLICATION TO THE GLOBAL ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3279-3306. doi: 10.11948/20250266

AN ALTERNATIVE RIEMANN-HILBERT APPROACH AND ITS APPLICATION TO THE GLOBAL ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS

1.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China
2.
School of Mathematics and Computer Science, Ningxia Normal University, Guyuan 756099, China
3.
Ningxia Key Laboratory of Artificial Intelligence and Basic Education, Ningxia Normal University, Guyuan 756099, China
4.
Institute for Computing and Information Sciences, University of Radboud, the Netherlands
5.
School of Science, Xi’an Shiyou University, Xi’an 710065, China

Author Bio: Email: 18795390879@163.com(J. Yan); Email: kumin0844@163.com(M. Ku); Email: xwei@xsyu.edu.cn(X. Wei)
Corresponding author: Email: nxhan@126.com(H. Han)

Abstract

Abstract

This work develops a direct solution method for a class of matrix Riemann-Hilbert (MRH) problems, avoiding the conventional reduction to scalar formulations. By leveraging the nonlinear steepest descent framework introduced by Deift and Zhou, we construct three invertible transformations that systematically reduce the MRH problem for orthogonal polynomials to an explicitly solvable model. Through analytic continuation and a rigorous treatment of boundary conditions, we obtain precise asymptotic expansions in both oscillatory and exponentially decaying regimes. Furthermore, our approach yields global asymptotics for orthogonal polynomials on extended complex domains, surpassing existing results confined to the unit circle or the real axis. These findings demonstrate the efficiency of our alternative Riemann-Hilbert approach and significantly broaden the utility of such techniques in the asymptotic analysis of orthogonal polynomials.
- Alternative Riemann-Hilbert approach /
- global asymptotic analysis /
- orthogonal polynomials with analytic weight
MSC: 30E25, 45E05, 42C05

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References

[1]	G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999. Google Scholar
[2]	A. I. Aptekarev, Sharp constants for rational approximations of analytic functions, Sb. Math., 2002, 193(1), 1–72. doi: 10.1070/SM2002v193n01ABEH000619 CrossRef Google Scholar
[3]	A. I. Aptekarev, S. Yu. Dobrokhotov, D. N. Tulyakov and A. V. Tsvetkova, Plancherel-Rotach type asymptotic formulae for multiple orthogonal Hermite polynomials and recurrence relations, Izv. Math., 2022, 86(1), 36–97. Google Scholar
[4]	C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers Ⅰ: Asymptotic Methods and Perturbation Theory, Springer, New York, 1999. Google Scholar
[5]	P. Bleher and K. Liechty, Uniform asymptotics for discrete orthogonal polynomials with respect to varying exponential weights on a regular infinite lattice, Int. Math. Res. Not., 2011, 2011(2), 342–386. Google Scholar
[6]	T. S. Chihara, An Introduction to Orthogonal Polynomials, Dover Publications, New York, 2011. Google Scholar
[7]	P. Deift, A. R. Its and X. Zhou, A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics, Ann. Math., 1997, 146, 26–47. Google Scholar
[8]	P. Deift and X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, asymptotics for the MKdV equation, Ann. Math., 1993, 137, 295–368. doi: 10.2307/2946540 CrossRef Google Scholar
[9]	X. C. Ding and T. Trogdon, A Riemann-Hilbert approach to the perturbation theory for orthogonal polynomials: Applications to numerical linear algebra and random matrix theory, Int. Math. Res. Not., 2024, 2024(5), 3975–4061. doi: 10.1093/imrn/rnad142 CrossRef Google Scholar
[10]	Z. H. Du and J. Y. Du, Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on the unit circle, Chin. Ann. Math. Ser. A, 2006, 27(5), 701–718. Google Scholar
[11]	A. Martinez-Finkelshtein, K. T. -R. McLaughlin and E. B. Saff, Szegö orthogonal polynomials with respect to an analytic weight: Canonical representation and strong asymptotics, Constr. Approx., 2006, 24, 319–363. doi: 10.1007/s00365-005-0617-6 CrossRef Google Scholar
[12]	P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, 2009. Google Scholar
[13]	A. S. Fokas, A. R. Its and A. V. Kitaev, The isomonodromy approach to matic models in 2D quantum gravity, Comm. Math. Phys., 1992, 147, 395–430. doi: 10.1007/BF02096594 CrossRef Google Scholar
[14]	Y. L. Geronimus, Orthogonal Polynomials: Estimates, Asymptotic Formulas, and Series of Polynomials Orthogonal on the Unit Circle and on an Interval, Authorized Translation from the Russian, Consultants Bureau, New York, 1961. Google Scholar
[15]	H. L. Han, H. Liu and Y. F. Wang, The strong asymptotic analysis of the first kind orthogonal trigonometric polynomial, preprint, 2021. arXiv:2103.03601v1 [math.CV]. Google Scholar
[16]	H. Hedenmalm, Soft Riemann-Hilbert problems and planar orthogonal polynomials, Comm. Pure Appl. Math., 2024, 77, 2413–2451. doi: 10.1002/cpa.22170 CrossRef Google Scholar
[17]	M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, 2005. Google Scholar
[18]	I. Karpenko, D. Shepelsky and G. Teschl, A Riemann-Hilbert approach to the modified Camassa-Holm equation with step-like boundary conditions, Monatsh. Math., 2023, 201, 127–172. doi: 10.1007/s00605-022-01786-y CrossRef Google Scholar
[19]	A. B. J. Kuijlaars, Riemann-Hilbert analysis for orthogonal polynomials, in: E. Koelink, W. Van Assche, eds., Orthogonal Polynomials and Special Functions, Lecture Notes in Mathematics, Vol. 1817, Springer, Berlin, Heidelberg, 2003, 167–210. Google Scholar
[20]	M. D. Liu and B. Li, Riemann-Hilbert approach for the complex Sharma-Tasso-Olver equation with high-order poles, Qual. Theory Dyn. Syst., 2024, 23, 225. doi: 10.1007/s12346-024-01053-5 CrossRef Google Scholar
[21]	J. K. Lu, Boundary Value Problems for Analytic Functions, World Scientific, Singapore, 1993. Google Scholar
[22]	P. Nevai and V. Totik, Orthogonal polynomials and their zeros, Acta Sci. Math., 1989, 53, 99–104. Google Scholar
[23]	P. L. Shabalin, The Riemann boundary value problem for generalized analytical functions with supersingular points on the contour of the boundary condition, Lobachevskii J. Math., 2024, 45, 5244–5253. doi: 10.1134/S1995080224605964 CrossRef Google Scholar
[24]	P. K. Suetin, Orthogonal Polynomials in Two Variables, Translated by E. V. Pankratiev, Routledge, London, 2022. Google Scholar
[25]	F. G. Tricomi, Sul comportamento asintotico dei polinomi di Laguerre, Ann. Mat. Pura Appl., 1949, 28(1), 263–289. doi: 10.1007/BF02411134 CrossRef Google Scholar
[26]	N. P. Vekua, Systems of Singular Integral Equations and Some Boundary Problems, Noordhoff, Groningen, 1967. Google Scholar
[27]	Y. F. Wang, Y. F. Lu and J. Y. Du, Strong asymptotic analysis of OLPs on the unit circle by Riemann- Hilbert approach, in: S. Rogosin and A. Çelebi, Analysis as a Life: Dedicated to Heinrich Begehr on the Occasion of his 80th Birthday, Birkhauser, Cham, 2019, 139–169. Google Scholar
[28]	R. Wong, Asymptotic Approximations of Integrals, Society for Industrial and Applied Mathematics, Philadelphia, 2001. Google Scholar