Linear difference equations with transition points
Authors:
Z. Wang and R. Wong
Journal:
Math. Comp. 74 (2005), 629653
MSC (2000):
Primary 41A60, 39A10, 33C45
Published electronically:
May 25, 2004
MathSciNet review:
2114641
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Two linearly independent asymptotic solutions are constructed for the secondorder linear difference equation
where and have power series expansions of the form with . Our results hold uniformly for in an infinite interval containing the transition point given by . As an illustration, we present an asymptotic expansion for the monic polynomials which are orthogonal with respect to the modified Jacobi weight , , where , and is real analytic and strictly positive on .
 1.
G. D. Birkhoff, Formal theory of irregular linear difference equations, Acta Math., 54 (1930), pp. 205246.
 2.
G. D. Birkhoff and W. J. Trjitzinsky, Analytic theory of singular difference equations, Acta Math., 60 (1932), pp. 189.
 3.
Pavel
Bleher and Alexander
Its, Semiclassical asymptotics of orthogonal polynomials,
RiemannHilbert problem, and universality in the matrix model, Ann. of
Math. (2) 150 (1999), no. 1, 185–266. MR 1715324
(2000k:42033), http://dx.doi.org/10.2307/121101
 4.
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
(2001g:42050), http://dx.doi.org/10.1002/(SICI)10970312(199911)52:11<1335::AIDCPA1>3.0.CO;21
 5.
P.
Deift, T.
Kriecherbauer, K.
TR 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
(2001f:42037), http://dx.doi.org/10.1002/(SICI)10970312(199912)52:12<1491::AIDCPA2>3.3.CO;2R
 6.
P.
Deift, T.
Kriecherbauer, K.
T.R. McLaughlin, S.
Venakides, and X.
Zhou, A RiemannHilbert 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
(2002h:42044), http://dx.doi.org/10.1016/S03770427(00)006348
 7.
P.
Deift and X.
Zhou, A steepest descent method for oscillatory RiemannHilbert
problems. Asymptotics for the MKdV equation, Ann. of Math. (2)
137 (1993), no. 2, 295–368. MR 1207209
(94d:35143), http://dx.doi.org/10.2307/2946540
 8.
R.
B. Dingle and G.
J. Morgan, 𝑊𝐾𝐵 methods for difference
equations. I, II, Appl. Sci. Res. 18 (1967/1968),
221–237; 238–245. MR 0225511
(37 #1104)
 9.
R.
B. Dingle and G.
J. Morgan, 𝑊𝐾𝐵 methods for difference
equations. I, II, Appl. Sci. Res. 18 (1967/1968),
221–237; 238–245. MR 0225511
(37 #1104)
 10.
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
(2000h:33016), http://dx.doi.org/10.1155/S1073792899000161
 11.
Arno
B. J. Kuijlaars and Kenneth
T.R. McLaughlin, RiemannHilbert analysis for Laguerre polynomials
with large negative parameter, Comput. Methods Funct. Theory
1 (2001), no. 1, 205–233. MR 1931612
(2003k:30059), http://dx.doi.org/10.1007/BF03320986
 12.
A. B. J. Kuijlaars, K. TR McLaughlin, W. Van Assche, and M. Vanlessen, The RiemannHilbert approach to strong asymptotics for orthogonal polynomials on , preprint math. CA/0111252.
 13.
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
(2003g:30043), http://dx.doi.org/10.1155/S1073792802203116
 14.
X.
Li and R.
Wong, On the asymptotics of the MeixnerPollaczek polynomials and
their zeros, Constr. Approx. 17 (2001), no. 1,
59–90. MR
1794802 (2001m:33012), http://dx.doi.org/10.1007/s003650010009
 15.
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
(97i:41001)
 16.
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 (electronic). MR 1759196
(2001f:42039), http://dx.doi.org/10.1137/S0036141098344671
 17.
Gábor
Szegő, Orthogonal polynomials, 4th ed., American
Mathematical Society, Providence, R.I., 1975. American Mathematical
Society, Colloquium Publications, Vol. XXIII. MR 0372517
(51 #8724)
 18.
W. Van Assche, J. S. Geronimo, and A. B. J. Kuijlaars, RiemannHilbert problems for multiple orthogonal polynomials, pp. 2350 in ``NATO ASI Special Functions 2000'' (J. Bustoz et. al. eds.), Kluwer Academic Publisher, Dordrecht, 2001.
 19.
Z.
Wang and R.
Wong, Uniform asymptotic expansion of
𝐽_{𝜈}(𝜈𝑎) via a difference equation,
Numer. Math. 91 (2002), no. 1, 147–193. MR 1896091
(2003g:33008), http://dx.doi.org/10.1007/s002110100316
 20.
Z. Wang and R. Wong, Asymptotic expansions for secondorder linear difference equations with a turning point, Numer. Math., 94 (2003), pp. 147194.
 21.
R.
Wong and H.
Li, Asymptotic expansions for secondorder linear difference
equations, J. Comput. Appl. Math. 41 (1992),
no. 12, 65–94. Asymptotic methods in analysis and
combinatorics. MR 1181710
(94i:39003), http://dx.doi.org/10.1016/03770427(92)90239T
 22.
R.
Wong and H.
Li, Asymptotic expansions for secondorder linear difference
equations. II, Stud. Appl. Math. 87 (1992),
no. 4, 289–324. MR 1182142
(94i:39004)
 1.
 G. D. Birkhoff, Formal theory of irregular linear difference equations, Acta Math., 54 (1930), pp. 205246.
 2.
 G. D. Birkhoff and W. J. Trjitzinsky, Analytic theory of singular difference equations, Acta Math., 60 (1932), pp. 189.
 3.
 P. Bleher and A. Its, Semiclassical asymptotics of orthogonal polynomials, RiemannHilbert problem, and universality in the matrix model, Ann. Math., 150 (1999), pp. 185266. MR 2000k:42033
 4.
 P. Deift, T. Kriecherbauer, K. TR 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), pp. 13351425. MR 2001g:42050
 5.
 P. Deift, T. Kriecherbauer, K. TR McLaughlin, S. Venakides, and X. Zhou, Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math., 52 (1999), pp. 14911552. MR 2001f:42037
 6.
 P. Deift, T. Kriecherbauer, K. TR McLaughlin, S. Venakides, and X. Zhou, A RiemannHilbert approach to asymptotic questions for orthogonal polynomials, J. Comput. Appl. Math., 133 (2001), pp. 4763. MR 2002h:42044
 7.
 P. Deift and X. Zhou, A steepest descent method for oscillatory RiemannHilbert problems, Applications for the MKdV equation, Ann. Math., 137 (1993), pp. 295368. MR 94d:35143
 8.
 R. B. Dingle and G. J. Morgan, WKB methods for difference equations I, Appl. Sci. Res., 18 (1967), pp. 221237. MR 37:1104
 9.
 R. B. Dingle and G. J. Morgan, WKB methods for difference equations II, Appl. Sci. Res., 18 (1967), pp. 238245. MR 37:1104
 10.
 T. Kriecherbauer and K. TR McLaughlin, Strong asymptotics of polynomials orthogonal with respect to Freud weights, Internat. Math. Res. Notices (1999), pp. 299333. MR 2000h:33016
 11.
 A. B. J. Kuijlaars and K. TR McLaughlin, RiemannHilbert analysis for Laguerre polynomials with large negative parameter, Comput. Meth. Funct. Theory, 1 (2001), pp. 205233. MR 2003k:30059
 12.
 A. B. J. Kuijlaars, K. TR McLaughlin, W. Van Assche, and M. Vanlessen, The RiemannHilbert approach to strong asymptotics for orthogonal polynomials on , preprint math. CA/0111252.
 13.
 A. B. J. Kuijlaars and M. Vanlessen, Universality for eigenvalue correlations from the modified Jacobi unitary ensemble, Internat. Math. Res. Notices (2002), pp. 15751600.MR 2003g:30043
 14.
 X. Li and R. Wong, On the asymptotics of the MeixnerPollaczek polynomials and their zeros, Constr. Approx., 17 (2001), pp. 5990.MR 2001m:33012
 15.
 F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. (Reprinted by A. K. Peters Ltd., Wellesley, 1997.) MR 97i:41001
 16.
 W.Y. Qiu and R. Wong, Uniform asymptotic formula for orthogonal polynomials with exponential weight, SIAM J. Math. Anal., 31 (2000), pp. 9921029. MR 2001f:42039
 17.
 G. Szegö, ``Orthogonal Polynomials," Fourth edition, Colloquium Publications, Vol. 23, Amer. Math. Soc. Providence R. I., 1975. MR 51:8724
 18.
 W. Van Assche, J. S. Geronimo, and A. B. J. Kuijlaars, RiemannHilbert problems for multiple orthogonal polynomials, pp. 2350 in ``NATO ASI Special Functions 2000'' (J. Bustoz et. al. eds.), Kluwer Academic Publisher, Dordrecht, 2001.
 19.
 Z. Wang and R. Wong, Uniform asymptotic expansion of via a difference equation, Numer. Math., 91 (2002), pp. 147193. MR 2003g:33008
 20.
 Z. Wang and R. Wong, Asymptotic expansions for secondorder linear difference equations with a turning point, Numer. Math., 94 (2003), pp. 147194.
 21.
 R. Wong and H. Li, Asymptotic expansions for secondorder linear difference equations, J. Comput. Appl. Math., 41 (1992), pp. 6594. MR 94i:39003
 22.
 R. Wong and H. Li, Asymptotic expansions for secondorder linear difference equations II, Stud. Appl. Math., 87 (1992), pp. 289324. MR 94i:39004
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC (2000):
41A60,
39A10,
33C45
Retrieve articles in all journals
with MSC (2000):
41A60,
39A10,
33C45
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
Email:
mawong@cityu.edu.hk
DOI:
http://dx.doi.org/10.1090/S0025571804016771
PII:
S 00255718(04)016771
Keywords:
Difference equation,
transition points,
threeterm recurrence relation,
orthogonal polynomials
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
Article copyright:
© Copyright 2004
American Mathematical Society
