{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T03:50:30Z","timestamp":1775620230698,"version":"3.50.1"},"reference-count":49,"publisher":"Springer Science and Business Media LLC","issue":"1-4","license":[{"start":{"date-parts":[[2008,2,6]],"date-time":"2008-02-06T00:00:00Z","timestamp":1202256000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2008,12]]},"DOI":"10.1007\/s11075-008-9159-x","type":"journal-article","created":{"date-parts":[[2008,2,5]],"date-time":"2008-02-05T10:23:53Z","timestamp":1202207033000},"page":"331-372","source":"Crossref","is-referenced-by-count":9,"title":["Global asymptotic expansions of the Laguerre polynomials\u2014a Riemann\u2013Hilbert approach"],"prefix":"10.1007","volume":"49","author":[{"given":"W.-Y.","family":"Qiu","sequence":"first","affiliation":[]},{"given":"R.","family":"Wong","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2008,2,6]]},"reference":[{"key":"9159_CR1","volume-title":"Handbook of Mathematical Functions","author":"M. Abramowitz","year":"1972","unstructured":"Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publications, New York (1972)"},{"key":"9159_CR2","series-title":"Annals of Mathematics Studies","volume-title":"Discrete Orthogonal Polynomials, Asymptotics and Applications","author":"J. Baik","year":"2007","unstructured":"Baik, J., Kricherbauer, T., McLaughlin, K.T.-R., Miller, P.D.: Discrete Orthogonal Polynomials, Asymptotics and Applications. Annals of Mathematics Studies, vol. 164. Princeton University Press, Princeton, NJ (2007)"},{"key":"9159_CR3","doi-asserted-by":"crossref","first-page":"185","DOI":"10.2307\/121101","volume":"150","author":"P. Bleher","year":"1999","unstructured":"Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann\u2013Hilbert problem, and universality in the matrix model. Ann. Math. 150, 185\u2013266 (1999)","journal-title":"Ann. Math."},{"key":"9159_CR4","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1002\/cpa.10065","volume":"56","author":"P. Bleher","year":"2003","unstructured":"Bleher, P., Its, A.: Double scaling limit in the random matrix model: the Riemann\u2013Hilbert approach. Comm. Pure Appl. Math. 56, 433\u2013516 (2003)","journal-title":"Comm. Pure Appl. Math."},{"key":"9159_CR5","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1017\/S0305004100032655","volume":"53","author":"C. Chester","year":"1957","unstructured":"Chester, C., Friedman, B., Ursell, F.: An extension of the method of steepest descents. Proc. Camb. Phil. Soc. 53, 599\u2013611 (1957)","journal-title":"Proc. Camb. Phil. Soc."},{"key":"9159_CR6","volume-title":"Cambridge Tracts in Math. and Math. Phys. No.55","author":"E.T. Copson","year":"1965","unstructured":"Copson, E.T.: Asymptotic Expansions. Cambridge Tracts in Math. and Math. Phys. No.55, Cambridge University Press, London (1965)"},{"key":"9159_CR7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11401-006-0195-3","volume":"28","author":"D. Dai","year":"2007","unstructured":"Dai, D., Wong, R.: Global asymptotics of Krawtchouk polynomials\u2014a Riemann\u2013Hilbert approach. Chin. Ann. Math. Ser. B 28, 1\u201334 (2007)","journal-title":"Chin. Ann. Math. Ser. B"},{"key":"9159_CR8","doi-asserted-by":"crossref","unstructured":"Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann\u2013Hilbert Approach. Courant Lecture Notes in Mathematics, No. 3., New York University (1999)","DOI":"10.1090\/cln\/003"},{"issue":"16","key":"9159_CR9","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1155\/S1073792897000500","volume":"1997","author":"P. Deift","year":"1997","unstructured":"Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R.: Asymptotics for polynomials orthogonal with respect to varying exponential weights. Int. Math. Res. Notices 1997(16), 759\u2013782 (1997)","journal-title":"Int. Math. Res. Notices"},{"key":"9159_CR10","doi-asserted-by":"crossref","first-page":"1335","DOI":"10.1002\/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1","volume":"52","author":"P. Deift","year":"1999","unstructured":"Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: 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, 1335\u20131425 (1999)","journal-title":"Comm. Pure Appl. Math."},{"key":"9159_CR11","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1002\/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.0.CO;2-#","volume":"52","author":"P. Deift","year":"1999","unstructured":"Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Comm. Pure Appl. Math. 52, 1491\u20131552 (1999)","journal-title":"Comm. Pure Appl. Math."},{"issue":"2","key":"9159_CR12","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1073\/pnas.95.2.450","volume":"95","author":"P. Deift","year":"1998","unstructured":"Deift, P., Venakides, S., Zhou, X.: An extension of the steepest descent method for Riemann\u2013Hilbert problems: the small dispersion limit of the Korteweg-de Vries (KdV) equation. Proc. Natl. Acad. Sci. USA 95(2), 450\u2013454 (1998)","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"9159_CR13","doi-asserted-by":"crossref","first-page":"295","DOI":"10.2307\/2946540","volume":"137","author":"P. Deift","year":"1993","unstructured":"Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann\u2013Hilbert problems. Asymptotics for the mKdV equation. Ann. Math. 137, 295\u2013370 (1993)","journal-title":"Ann. Math."},{"key":"9159_CR14","unstructured":"Deift, P., Zhou, X.: Long-time behavior of the non-focusing nonlinear Schr\u00f6dinger equation: a case study. New Series: Lectures in Math. Sciences, vol. 5. University of Tokyo (1994)"},{"key":"9159_CR15","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1002\/cpa.3160480304","volume":"48","author":"P Deift","year":"1995","unstructured":"Deift, P, Zhou, X., Asymptotics for the Painlev\u00e9 II equation. Comm. Pure Appl. Math. 48, 277\u2013337 (1995)","journal-title":"Comm. Pure Appl. Math."},{"key":"9159_CR16","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1007\/BF02096594","volume":"147","author":"A.S. Fokas","year":"1992","unstructured":"Fokas, A.S., Its, A.R., Kitaev, A.V.: The isomonodromy approach to matrix models in 2D quantum gravity. Comm. Math. Phys. 147, 395\u2013430 (1992)","journal-title":"Comm. Math. Phys."},{"key":"9159_CR17","doi-asserted-by":"crossref","first-page":"1232","DOI":"10.1137\/0519087","volume":"19","author":"C.L. Frenzen","year":"1988","unstructured":"Frenzen, C.L., Wong, R.: Uniform asymptotic expansions of Laguerre polynomials. SIAM J. Math. Anal. 19, 1232\u20131248 (1988)","journal-title":"SIAM J. Math. Anal."},{"key":"9159_CR18","doi-asserted-by":"crossref","unstructured":"Gatteschi, L.: Uniform approximations for the zeros of Laguerre polynomials. In: Numerical Mathematics, Proceedings of the International Conference, Singapore 1988. Int. Ser. Numer. Math. 86, 137\u2013148 (1988)","DOI":"10.1007\/978-3-0348-6303-2_12"},{"key":"9159_CR19","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1016\/S0377-0427(01)00549-0","volume":"144","author":"L. Gatteschi","year":"2002","unstructured":"Gatteschi, L.: Asymptotics and bounds for the zeros of Laguerre polynomials: a survey. J. Comput. Appl. Math. 144, 7\u201327 (2002)","journal-title":"J. Comput. Appl. Math."},{"key":"9159_CR20","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1216\/rmjm\/1181069964","volume":"33","author":"W. Gawronski","year":"2003","unstructured":"Gawronski, W., Van Assche, W.: Strong asymptotics for relativistic Hermite polynomials. Rocky Mountain J. Math. 33, 489\u2013524 (2003)","journal-title":"Rocky Mountain J. Math."},{"key":"9159_CR21","doi-asserted-by":"crossref","unstructured":"Its, A.R., Kapaev, A.A.: The nonlinear steepest descent approach to the asymptotics of the second Painlev\u00e9 transcendent in the complex domain, Phys. Odyssey 2001. Integrable Models and Beyond (2002)","DOI":"10.1007\/978-1-4612-0087-1_10"},{"issue":"6","key":"9159_CR22","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1155\/S1073792899000161","volume":"1999","author":"T. Kricherbauer","year":"1999","unstructured":"Kricherbauer, T., McLaughlin, K.T.-R.: Strong asymptotics of polynomials orthogonal with respect to Freud weights. Int. Math. Res. Notices 1999(6), 299\u2013333 (1999)","journal-title":"Int. Math. Res. Notices"},{"key":"9159_CR23","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/3-540-44945-0_5","volume-title":"Orthogonal Polynomials and Special Functions: Leuven 2002 Lecture Notes in Mathematics, vol. 1817","author":"A.B.J. Kuijlaars","year":"2003","unstructured":"Kuijlaars, A.B.J.: Riemann\u2013Hilbert analysis for orthogonal polynomials. In: Koelink, E., Van Assche, W. (eds.) Orthogonal Polynomials and Special Functions: Leuven 2002 Lecture Notes in Mathematics, vol. 1817, pp.\u00a0167\u2013210. Springer-Verlag, Berlin (2003)"},{"key":"9159_CR24","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/BF02789047","volume":"94","author":"A.B.J. Kuijlaars","year":"2004","unstructured":"Kuijlaars, A.B.J., Martinez-Finkelshtein, A.: Strong asymptotics for Jacobi polynomials with varying nonstandard parameters. J. Anal. Math. 94, 195\u2013234 (2004)","journal-title":"J. Anal. Math."},{"key":"9159_CR25","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1007\/BF03320986","volume":"1","author":"A.B.J. Kuijlaars","year":"2001","unstructured":"Kuijlaars, A.B.J., McLaughlin, K.T.-R.: Riemann\u2013Hilbert analysis for Laguerre polynomials with large negative parameter. Comput. Methods Funct. Theory 1, 205\u2013233 (2001)","journal-title":"Comput. Methods Funct. Theory"},{"key":"9159_CR26","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1007\/s00365-003-0536-3","volume":"20","author":"A.B.J. Kuijlaars","year":"2004","unstructured":"Kuijlaars, A.B.J., McLaughlin, K.T.-R.: Asymptotic zero behavior of Laguerre polynomials with negative parameter. Constr. Approx. 20, 497\u2013523 (2004)","journal-title":"Constr. Approx."},{"key":"9159_CR27","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/j.cam.2004.01.043","volume":"178","author":"A.B.J. Kuijlaars","year":"2005","unstructured":"Kuijlaars, A.B.J., McLaughlin, K.T.-R.: A Riemann\u2013Hilbert problem for biorthogonal polynomials. J. Comput. Appl. Math. 178, 313\u2013320 (2005)","journal-title":"J. Comput. Appl. Math."},{"key":"9159_CR28","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/j.aim.2003.08.015","volume":"188","author":"A.B.J. Kuijlaars","year":"2004","unstructured":"Kuijlaars, A.B.J., McLaughlin, K.T.-R., Van Assche W., Vanlessen, M.: The Riemann\u2013Hilbert approach to strong asymptotics for orthogonal polynomials on [\u22121,1]. Adv. Math. 188, 337\u2013398 (2004)","journal-title":"Adv. Math."},{"key":"9159_CR29","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/s00365-005-0617-6","volume":"24","author":"A. Martinez-Finkelshtein","year":"2006","unstructured":"Martinez-Finkelshtein, A., McLaughlin, K.T.-R., Saff, E.B.: Szeg\u00f6 orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics. Constr. Approx. 24, 319\u2013363 (2006)","journal-title":"Constr. Approx."},{"key":"9159_CR30","unstructured":"Martinez-Finkelshtein, A., McLaughlin, K.T.-R., Saff, E.B.: Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle. Int. Math. Res. Notices 2006, Art. ID 91426, 1\u201343 (2006)"},{"key":"9159_CR31","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.jat.2005.02.004","volume":"134","author":"A. Martinez-Finkelshtein","year":"2005","unstructured":"Martinez-Finkelshtein, A., Orive, R.: Riemann\u2013Hilbert analysis of Jacobi polynomials orthogonal on a single contour. J. Approx. Theory 134, 137-170 (2005)","journal-title":"J. Approx. Theory"},{"key":"9159_CR32","volume-title":"Singular Integral Equations","author":"N.I. Muskhelishvili","year":"1992","unstructured":"Muskhelishvili, N.I.: Singular Integral Equations, 2nd ed. Dover Publ. Inc., New York (1992)","edition":"2"},{"key":"9159_CR33","volume-title":"Asymptotics and Special Functions","author":"F.W.J. Olver","year":"1974","unstructured":"Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974) (reprinted by Peters, A.K., Ltd., Wellesley 1997)"},{"key":"9159_CR34","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF03321065","volume":"4","author":"W.-Y. Qiu","year":"2004","unstructured":"Qiu W.-Y., Wong, R.: Asymptotic expansion of the Krawchouk polynomials and their zeros. Comput. Method Function Theory 4, 189\u2013226 (2004)","journal-title":"Comput. Method Function Theory"},{"key":"9159_CR35","unstructured":"Qiu W.-Y., Wong, R.: Riemann\u2013Hilbert Method in Asymptotic Analysis of Orthogonal Polynomials, Analysis and Applications (2008) (to appear)"},{"key":"9159_CR36","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03329-6","volume-title":"Logarithmic Potentials with External Fields","author":"E.B. Saff","year":"1997","unstructured":"Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Springer-Verlag, Berlin (1997)"},{"key":"9159_CR37","series-title":"Colloquium Publications","doi-asserted-by":"crossref","DOI":"10.1090\/coll\/023","volume-title":"Orthogonal polynomials","author":"G. Szeg\u00f6","year":"1939","unstructured":"Szeg\u00f6, G.: Orthogonal polynomials. Colloquium Publications, vol. 23. Amer. Math. Soc., Providence, R.I. (1939)"},{"key":"9159_CR38","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s00365-005-0611-z","volume":"25","author":"M. Vanlessen","year":"2007","unstructured":"Vanlessen, M.: Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory. Constr. Approx. 25, 125\u2013175 (2007)","journal-title":"Constr. Approx."},{"key":"9159_CR39","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s002110100316","volume":"91","author":"Z. Wang","year":"2002","unstructured":"Wang, Z., Wong, R.: Uniform asymptotic expansion of J \u03bd(\u03bda) via a difference equation. Numer. Math. 91, 147\u2013193 (2002)","journal-title":"Numer. Math."},{"key":"9159_CR40","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s00211-002-0416-y","volume":"94","author":"Z. Wang","year":"2003","unstructured":"Wang, Z., Wong, R.: Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94, 147\u2013194 (2003)","journal-title":"Numer. Math."},{"key":"9159_CR41","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1090\/S0025-5718-04-01677-1","volume":"74","author":"Z. Wang","year":"2005","unstructured":"Wang, Z., Wong, R.: Linear difference equations with transition points. Math. Comp. 74, 629\u2013653 (2005)","journal-title":"Math. Comp."},{"key":"9159_CR42","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1111\/j.1467-9590.2005.01567","volume":"115","author":"Z. Wang","year":"2005","unstructured":"Wang, Z., Wong, R., Uniform asymptotics for orthogonal polynomials with exponential weights\u2014the Riemann\u2013Hilbert approach. Stud. Appl. Math. 115, 139\u2013155 (2005)","journal-title":"Stud. Appl. Math."},{"key":"9159_CR43","doi-asserted-by":"crossref","first-page":"2839","DOI":"10.1098\/rspa.2005.1467","volume":"461","author":"Z. Wang","year":"2005","unstructured":"Wang, Z., Wong, R.: Bessel-type asymptotic expansions via the Riemann\u2013Hilbert approach. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461, 2839\u20132856 (2005)","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci."},{"issue":"9","key":"9159_CR44","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1016\/j.matpur.2005.11.005","volume":"85","author":"Z. Wang","year":"2006","unstructured":"Wang, Z., Wong, R.: Uniform asymptotics of the Stieltjes\u2013Wigert polynomials via the Riemann\u2013Hilbert approach. J. Math. Pures Appl. 85(9), 698\u2013718 (2006)","journal-title":"J. Math. Pures Appl."},{"key":"9159_CR45","volume-title":"Asymptotic Approximations of Integrals","author":"R. Wong","year":"1989","unstructured":"Wong, R.: Asymptotic Approximations of Integrals. Academic Press, Boston (1989) (reprinted by SIAM, Philadelphia, 2001)."},{"key":"9159_CR46","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/0377-0427(92)90239-T","volume":"41","author":"R. Wong","year":"1992","unstructured":"Wong, R., Li, H.: Asymptotic expansions for second order linear difference equations. J. Comput. Appl. Math. 41, 65\u201394 (1992)","journal-title":"J. Comput. Appl. Math."},{"key":"9159_CR47","doi-asserted-by":"crossref","first-page":"661","DOI":"10.3934\/dcdsb.2007.7.661","volume":"7","author":"R. Wong","year":"2007","unstructured":"Wong, R., Zhang, L.: Global asymptotics of Hermite polynomials via Riemann\u2013Hilbert approach. Discrete Contin. Dyn. Syst. Ser. B 7, 661\u2013682 (2007)","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"9159_CR48","doi-asserted-by":"crossref","first-page":"2663","DOI":"10.1090\/S0002-9947-06-03901-8","volume":"358","author":"R. Wong","year":"2006","unstructured":"Wong, R., Zhang, W.-J.: Uniform asymptotics for Jacobi polynomials with varying large negative parameters\u2014a Riemann\u2013Hilbert approach. Trans. Amer. Math. Soc. 358, 2663\u20132694 (2006)","journal-title":"Trans. Amer. Math. Soc."},{"key":"9159_CR49","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1142\/S0219530503000107","volume":"1","author":"R. Wong","year":"2003","unstructured":"Wong, R., Zhao, Y.-Q.: Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials. Anal. Appl. 1, 213\u2013241 (2003)","journal-title":"Anal. Appl."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-008-9159-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s11075-008-9159-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-008-9159-x","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,1]],"date-time":"2019-06-01T05:08:10Z","timestamp":1559365690000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s11075-008-9159-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,2,6]]},"references-count":49,"journal-issue":{"issue":"1-4","published-print":{"date-parts":[[2008,12]]}},"alternative-id":["9159"],"URL":"https:\/\/doi.org\/10.1007\/s11075-008-9159-x","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,2,6]]}}}