Transactions of the American Mathematical Society

Author(s): E. Coussement; J. Coussement; W. Van Assche
Journal: Trans. Amer. Math. Soc. 360 (2008), 5571-5588.
MSC (2000): Primary 33C45, 42C05; Secondary 15A18
Posted: May 20, 2008
Retrieve article in: PDF

Abstract: We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples of multiple orthogonal polynomials, in particular Jacobi-Piñeiro, Laguerre I and the example associated with modified Bessel functions. We also discuss an application to Toeplitz matrices.

1.: M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover Publications, New York, 1968.
2.: A. I. Aptekarev. Multiple orthogonal polynomials. J. Comput. Appl. Math. 99 (1998), 423-447. MR 1662713 (99m:42036)
3.: A. I. Aptekarev, P. M. Bleher, and A. B. J. Kuijlaars. Large limit of Gaussian random matrices with external source, part II. Comm. Math. Phys. 259 (2005), 367-389. MR 2172687 (2006f:82036)
4.: A. I. Aptekarev, V. Kalyagin, G. López Lagomasino, and I. A. Rocha. On the limit behavior of recurrence coefficients for multiple orthogonal polynomials. J. Approx. Theory 139 (2006), 346-370. MR 2220045 (2007a:42048)
5.: A. I. Aptekarev, V. Kaliaguine, and J. Van Iseghem. The genetic sums' representation for the moments of a system of Stieltjes functions and its application. Constr. Approx. 16 (2000), 487-524. MR 1771693 (2001g:41021)
6.: P. M. Bleher and A. B. J. Kuijlaars. Integral representations for multiple Hermite and multiple Laguerre polynomials. Annales de l'Institut Fourier 55 (2005), 2001-2014. MR 2187942 (2007c:33014)
7.: P. M. Bleher and A. B. J. Kuijlaars. Random matrices with external source and multiple orthogonal polynomials. Internat. Math. Research Notices 2004:3 (2004), 109-129. MR 2038771 (2004k:82034)
8.: P. M. Bleher and A. B. J. Kuijlaars. Large limit of Gaussian random matrices with external source, part I. Comm. Math. Phys. 252 (2004), 43-76. MR 2103904 (2005i:82031)
9.: M. G. de Bruin. Simultaneous Padé approximation and orthogonality. In Orthogonal polynomials and applications (C. Brezinski et al., ed.), volume 1171 of Lecture Notes in Mathematics, pages 74-83. Springer, Berlin, 1985. MR 838972 (87m:41019)
10.: M. G. de Bruin. Some aspects of simultaneous rational approximation. In Numerical Analysis and Mathematical Modeling, volume 24 of Banach Center Publications, pages 51-84. PWN-Polish Scientific Publishers, Warsaw, 1990. MR 1097402 (92b:41030)
11.: A. Bultheel, A. Cuyt, M. Van Barel, B. Verdonk, and W. Van Assche. Generalizations of orthogonal polynomials. J. Comput. Appl. Math. 179 (2005), 57-95. MR 2134361 (2006a:42034)
12.: Y. Ben Cheikh and N. Ben Romdhane. -orthogonal polynomial sets of Chebyshev type. Preprint
13.: T. S. Chihara. An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978. MR 0481884 (58:1979)
14.: K. Douak and P. Maroni. On -orthogonal Tchebychev polynomials, I. Appl. Num. Math. 24 (1997), 23-53. MR 1454707 (98g:42038)
15.: F. P. Gantmacher and M. G. Krein. Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, Revised edition. AMS Chelsea Publishing, Providence, RI, 2002. MR 1908601 (2003f:34161)
16.: J. Geronimo and T. P. Hill. Necessary and sufficient condition that the limit of Stieltjes transforms is a Stieltjes transform. J. Approx. Theory 121 (2003), no. 1, 54-60. MR 1962995 (2004a:60040)
17.: P. Henrici. Special functions - integral transforms - asymptotics - continued fractions, volume 2 of Applied and computational complex analysis (reprint of the 1977 original). Wiley Classics Library, Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1991. MR 1164865 (93b:30001)
18.: M. E. H. Ismail. Classical and Quantum Orthogonal Polynomials in One Variable, volume 98 in Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge, 2005. MR 2191786 (2007f:33001)
19.: A. B. J. Kuijlaars. Chebyshev quadrature for measures with strong singularity. J. Comp. Appl. Math. 65 (1995), 207-214. MR 1379132 (96m:41050)
20.: A. B. J. Kuijlaars and S. Serra Capizzano. Asymptotic zero distribution of orthogonal polynomials with discontinuously varying recurrence coefficients. J. Approx. Theory 113 (2001), no. 1, 142-155. MR 1866252 (2002i:42029)
21.: A. B. J. Kuijlaars and W. Van Assche. The asymptotic zero distribution of orthogonal polynomials with varying recurrence coefficients. J. Appr. Theory 99 (1999), 167-197. MR 1696553 (2000h:42015)
22.: K. Mahler. Perfect systems. Compositio Math. 19 (1968), 95-166. MR 0239099 (39:458)
23.: N. I. Muskhelishvili. Singular integral equations, Boundary problems of function theory and their application to mathematical physics, Translated from the second (1946) Russian edition and with a preface by J. R. M. Radok. Corrected reprint of the 1953 English translation, Dover Publications, Inc., New York, 1992. MR 1215485 (94a:45001)
24.: P. Nevai. Orthogonal polynomials. Mem. Amer. Math. Soc. 213 (1979). MR 519926 (80k:42025)
25.: E. M. Nikishin and V. N. Sorokin. Rational Approximants and Orthogonality, volume 92 in Translations of Mathematical Monographs. Amer. Math. Soc., Providence, RI, 1991. MR 1130396 (92i:30037)
26.: E. B. Saff and V. Totik. Logarithmic Potentials with External Fields, volume 316 in Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin/NY, 1997. MR 1485778 (99h:31001)
27.: W. Van Assche. Multiple orthogonal polynomials, irrationality and transcendence. In volume 236 of Continued fractions: from Analytic Number Theory to Constructive Approximation, Contemporary Mathematics (B. C. Berndt et al., ed.), pages 325-342. Amer. Math. Soc., Providence RI, 1999. MR 1665377 (2000k:42039)
28.: W. Van Assche. Asymptotics for orthogonal polynomials and three-term recurrences. In ``Orthogonal Polynomials'' (P. Nevai, ed.), volume 294 of NATO ASI Series C, pages 435-462. Kluwer Academic, Dordrecht, 1990. MR 1100305 (92k:42038)
29.: W. Van Assche and E. Coussement. Some classical multiple orthogonal polynomials. J. Comput. Appl. Math. 127 (2001), 317-347. MR 1808581 (2001i:33012)
30.: W. Van Assche and S. B. Yakubovich. Multiple orthogonal polynomials associated with Macdonald functions. Integral Transforms Special Funct. 9 (2000), 229-244. MR 1782974 (2001e:42034)
31.: A. Wintner. Spektraltheorie der Unendlichen Matrizen. Hirzel, Leipzig, 1929.

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 33C45, 42C05, 15A18

E. Coussement
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium

J. Coussement
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium

W. Van Assche
Affiliation: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium
Email: walter@wis.kuleuven.be

DOI: 10.1090/S0002-9947-08-04535-2
PII: S 0002-9947(08)04535-2
Keywords: Multiple orthogonal polynomials, asymptotics
Received by editor(s): June 19, 2006
Received by editor(s) in revised form: January 31, 2007
Posted: May 20, 2008
Additional Notes: This work was supported by INTAS project 03-51-6637, by FWO projects G.0455.04 and G.0184.02 and by OT/04/21 of Katholieke Universiteit Leuven
The second author is a postdoctoral researcher at the Katholieke Universiteit Leuven (Belgium)
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS