Linear difference equations with transition points
HTML articles powered by AMS MathViewer
- by Z. Wang and R. Wong PDF
- Math. Comp. 74 (2005), 629-653 Request permission
Abstract:
Two linearly independent asymptotic solutions are constructed for the second-order linear difference equation \begin{equation} y_{n+1}(x)-(A_nx+B_n)y_n(x)+y_{n-1}(x)=0, \nonumber \end{equation} where $A_n$ and $B_n$ have power series expansions of the form \begin{equation} A_n\sim \sum ^\infty _{s=0}\frac {\alpha _s}{n^s}, \qquad \qquad B_n\sim \sum ^\infty _{s=0}\frac {\beta _s}{n^s}\nonumber \end{equation} with $\alpha _0\ne 0$. Our results hold uniformly for $x$ in an infinite interval containing the transition point $x_+$ given by $\alpha _0 x_++\beta _0=2$. As an illustration, we present an asymptotic expansion for the monic polynomials $\pi _n(x)$ which are orthogonal with respect to the modified Jacobi weight $w(x)=(1-x)^\alpha (1+x)^\beta h(x)$, $x\in (-1,1)$, where $\alpha$, $\beta >-1$ and $h$ is real analytic and strictly positive on $[-1, 1]$.References
- G. D. Birkhoff, Formal theory of irregular linear difference equations, Acta Math., 54 (1930), pp. 205-246.
- G. D. Birkhoff and W. J. Trjitzinsky, Analytic theory of singular difference equations, Acta Math., 60 (1932), pp. 1-89.
- Pavel Bleher and Alexander Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. (2) 150 (1999), no. 1, 185–266. MR 1715324, DOI 10.2307/121101
- P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math. 52 (1999), no. 11, 1335–1425. MR 1702716, DOI 10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1
- 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, DOI 10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.3.CO;2-R
- P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), 2001, pp. 47–63. MR 1858269, DOI 10.1016/S0377-0427(00)00634-8
- 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, DOI 10.2307/2946540
- R. B. Dingle and G. J. Morgan, $\textrm {WKB}$ methods for difference equations. I, II, Appl. Sci. Res. 18 (1967/68), 221–237; 238–245. MR 225511, DOI 10.1007/BF00382348
- R. B. Dingle and G. J. Morgan, $\textrm {WKB}$ methods for difference equations. I, II, Appl. Sci. Res. 18 (1967/68), 221–237; 238–245. MR 225511, DOI 10.1007/BF00382348
- T. Kriecherbauer and K. T.-R. McLaughlin, Strong asymptotics of polynomials orthogonal with respect to Freud weights, Internat. Math. Res. Notices 6 (1999), 299–333. MR 1680380, DOI 10.1155/S1073792899000161
- 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, [On table of contents: 2002], 205–233. MR 1931612, DOI 10.1007/BF03320986
- A. B. J. Kuijlaars, K. T-R McLaughlin, W. Van Assche, and M. Vanlessen, The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$, preprint math. CA/0111252.
- A. B. J. Kuijlaars and M. Vanlessen, Universality for eigenvalue correlations from the modified Jacobi unitary ensemble, Int. Math. Res. Not. 30 (2002), 1575–1600. MR 1912278, DOI 10.1155/S1073792802203116
- X. Li and R. Wong, On the asymptotics of the Meixner-Pollaczek polynomials and their zeros, Constr. Approx. 17 (2001), no. 1, 59–90. MR 1794802, DOI 10.1007/s003650010009
- 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, DOI 10.1201/9781439864548
- W.-Y. Qiu and R. Wong, Uniform asymptotic formula for orthogonal polynomials with exponential weight, SIAM J. Math. Anal. 31 (2000), no. 5, 992–1029. MR 1759196, DOI 10.1137/S0036141098344671
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
- W. Van Assche, J. S. Geronimo, and A. B. J. Kuijlaars, Riemann-Hilbert problems for multiple orthogonal polynomials, pp. 23-50 in “NATO ASI Special Functions 2000” (J. Bustoz et. al. eds.), Kluwer Academic Publisher, Dordrecht, 2001.
- Z. Wang and R. Wong, Uniform asymptotic expansion of $J_\nu (\nu a)$ via a difference equation, Numer. Math. 91 (2002), no. 1, 147–193. MR 1896091, DOI 10.1007/s002110100316
- Z. Wang and R. Wong, Asymptotic expansions for second-order linear difference equations with a turning point, Numer. Math., 94 (2003), pp. 147-194.
- R. Wong and H. Li, Asymptotic expansions for second-order linear difference equations, J. Comput. Appl. Math. 41 (1992), no. 1-2, 65–94. Asymptotic methods in analysis and combinatorics. MR 1181710, DOI 10.1016/0377-0427(92)90239-T
- R. Wong and H. Li, Asymptotic expansions for second-order linear difference equations. II, Stud. Appl. Math. 87 (1992), no. 4, 289–324. MR 1182142, DOI 10.1002/sapm1992874289
Additional Information
- Z. Wang
- Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, P. O. Box 71010, Wuhan 430071, Peoples Republic of China
- Email: mcwang@cityu.edu.hk
- R. Wong
- Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
- MR Author ID: 192744
- Email: mawong@cityu.edu.hk
- Received by editor(s): April 2, 2003
- Received by editor(s) in revised form: October 6, 2003
- Published electronically: May 25, 2004
- Additional Notes: The work of this author was partially supported by the Research Grant Council of Hong Kong under Project 9040522
- © Copyright 2004 American Mathematical Society
- Journal: Math. Comp. 74 (2005), 629-653
- MSC (2000): Primary 41A60, 39A10, 33C45
- DOI: https://doi.org/10.1090/S0025-5718-04-01677-1
- MathSciNet review: 2114641