Mehler-Heine asymptotics for multiple orthogonal polynomials
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Abstract:
Mehler-Heine asymptotics describe the behavior of orthogonal polynomials near the edges of the interval where the orthogonality measure is supported. For Jacobi polynomials and Laguerre polynomials this asymptotic behavior near the hard edge involves Bessel functions $J_\alpha$. We show that the asymptotic behavior near the endpoint of the interval of (one of) the measures for multiple orthogonal polynomials involves a generalization of the Bessel function. The multiple orthogonal polynomials considered are Jacobi-Angelesco polynomials, Jacobi-Piñeiro polynomials, multiple Laguerre polynomials, multiple orthogonal polynomials associated with modified Bessel functions (of the first and second kind), and multiple orthogonal polynomials associated with Meijer $G$-functions.References
- A. I. Aptekarev, A. Branquinho, and W. Van Assche, Multiple orthogonal polynomials for classical weights, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3887–3914. MR 1990569, DOI 10.1090/S0002-9947-03-03330-0
- Youssèf Ben Cheikh and Khalfa Douak, On two-orthogonal polynomials related to the Bateman’s $J_n^{u,v}$-function, Methods Appl. Anal. 7 (2000), no. 4, 641–662. MR 1868550, DOI 10.4310/MAA.2000.v7.n4.a3
- Els Coussement and Walter Van Assche, Some properties of multiple orthogonal polynomials associated with Macdonald functions, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), 2001, pp. 253–261. MR 1858284, DOI 10.1016/S0377-0427(00)00648-8
- Els Coussement and Walter Van Assche, Multiple orthogonal polynomials associated with the modified Bessel functions of the first kind, Constr. Approx. 19 (2003), no. 2, 237–263. MR 1957385, DOI 10.1007/s00365-002-0499-9
- Els Coussement and Walter Van Assche, Asymptotics of multiple orthogonal polynomials associated with the modified Bessel functions of the first kind, Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), 2003, pp. 141–149. MR 1985686, DOI 10.1016/S0377-0427(02)00596-4
- 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
- Klaas Deschout and Arno B. J. Kuijlaars, Double scaling limit for modified Jacobi-Angelesco polynomials, Notions of positivity and the geometry of polynomials, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2011, pp. 115–161. MR 3051164, DOI 10.1007/978-3-0348-0142-3_{8}
- K. Deschout and A. B. J. Kuijlaars, Critical behavior in Angelesco ensembles, J. Math. Phys. 53 (2012), no. 12, 123523, 21. MR 3405913, DOI 10.1063/1.4769822
- Khalfa Douak, On $2$-orthogonal polynomials of Laguerre type, Int. J. Math. Math. Sci. 22 (1999), no. 1, 29–48. MR 1684357, DOI 10.1155/S0161171299220297
- Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2009. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey; Reprint of the 2005 original. MR 2542683
- Arno B. J. Kuijlaars, Riemann-Hilbert analysis for orthogonal polynomials, Orthogonal polynomials and special functions (Leuven, 2002) Lecture Notes in Math., vol. 1817, Springer, Berlin, 2003, pp. 167–210. MR 2022855, DOI 10.1007/3-540-44945-0_{5}
- Arno B. J. Kuijlaars, Multiple orthogonal polynomial ensembles, Recent trends in orthogonal polynomials and approximation theory, Contemp. Math., vol. 507, Amer. Math. Soc., Providence, RI, 2010, pp. 155–176. MR 2647568, DOI 10.1090/conm/507/09958
- 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 $[-1,1]$, Adv. Math. 188 (2004), no. 2, 337–398. MR 2087231, DOI 10.1016/j.aim.2003.08.015
- Arno B. J. Kuijlaars and Lun Zhang, Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits, Comm. Math. Phys. 332 (2014), no. 2, 759–781. MR 3257662, DOI 10.1007/s00220-014-2064-3
- Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
- V. N. Sorokin, Generalization of classical orthogonal polynomials and convergence of simultaneous Padé approximants, Trudy Sem. Petrovsk. 11 (1986), 125–165, 245, 247 (Russian, with English summary); English transl., J. Soviet Math. 45 (1989), no. 6, 1461–1499. MR 834171, DOI 10.1007/BF01097274
- G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, Providence RI, 1939 (fourth edition, 1975, MR0372517).
- Tomohiro Takata, Asymptotic formulae of Mehler-Heine-type for certain classical polyorthogonal polynomials, J. Approx. Theory 135 (2005), no. 2, 160–175. MR 2158528, DOI 10.1016/j.jat.2005.04.005
- Tomohiro Takata, Certain multiple orthogonal polynomials and a discretization of the Bessel equation, J. Math. Kyoto Univ. 49 (2009), no. 4, 747–769. MR 2591115, DOI 10.1215/kjm/1265899481
- D. N. Tulyakov, Difference schemes with power-growth bases perturbed by the spectral parameter, Mat. Sb. 200 (2009), no. 5, 129–158 (Russian, with Russian summary); English transl., Sb. Math. 200 (2009), no. 5-6, 753781. MR 2541225, DOI 10.1070/SM2009v200n05ABEH004018
- Walter Van Assche, Jeffrey S. Geronimo, and Arno B. J. Kuijlaars, Riemann-Hilbert problems for multiple orthogonal polynomials, Special functions 2000: current perspective and future directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., vol. 30, Kluwer Acad. Publ., Dordrecht, 2001, pp. 23–59. MR 2006283, DOI 10.1007/978-94-010-0818-1_{2}
- W. Van Assche and S. B. Yakubovich, Multiple orthogonal polynomials associated with Macdonald functions, Integral Transform. Spec. Funct. 9 (2000), no. 3, 229–244. MR 1782974, DOI 10.1080/10652460008819257
- M. Vanlessen, Strong asymptotics of Laguerre-type orthogonal polynomials and applications in random matrix theory, Constr. Approx. 25 (2007), no. 2, 125–175. MR 2283495, DOI 10.1007/s00365-005-0611-z
- R. Wong and Y.-Q. Zhao, Smoothing of Stokes’s discontinuity for the generalized Bessel function, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1984, 1381–1400. MR 1701756, DOI 10.1098/rspa.1999.0365
- E. Maitland Wright, The Asymptotic Expansion of the Generalized Bessel Function, Proc. London Math. Soc. (2) 38 (1935), 257–270. MR 1576315, DOI 10.1112/plms/s2-38.1.257
Additional Information
- Walter Van Assche
- Affiliation: Department of Mathematics, KU Leuven, Celestijnenlaan 200B, Box 2400, BE 3001 Leuven, Belgium
- MR Author ID: 176825
- ORCID: 0000-0003-3446-6936
- Email: walter@wis.kuleuven.be
- Received by editor(s): August 26, 2014
- Received by editor(s) in revised form: March 20, 2016, and March 24, 2016
- Published electronically: July 12, 2016
- Additional Notes: This research was supported by KU Leuven Research Grant OT/12/073, FWO Research Grant G.0934.13 and the Belgian Interuniversity Attraction Poles Programme P7/18.
- Communicated by: Mourad Ismail
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 303-314
- MSC (2010): Primary 33C45, 42C05
- DOI: https://doi.org/10.1090/proc/13214
- MathSciNet review: 3565381