Integral representations for multiple Hermite and multiple Laguerre polynomials
Annales de l'Institut Fourier, Volume 55 (2005) no. 6, p. 2001-2014

We give integral representations for multiple Hermite and multiple Laguerre polynomials of both type I and II. We also show how these are connected with double integral representations of certain kernels from random matrix theory.

On présente des représentations intégrales doubles pour les polynômes d’Hermite et de Laguerre multiples, aussi bien ceux de type I que ceux de type II. On montre aussi la connexion avec les représentations intégrales de certains noyaux de la théorie des matrices aléatoires.

DOI : https://doi.org/10.5802/aif.2148
Classification:  42C05,  15A52
Keywords: Multiple orthogonal polynomials, random matrices, Christoffel-Darboux formula 
@article{AIF_2005__55_6_2001_0,
     author = {M. BLEHER, Pavel and B.J. Kuijlaars, Arno},
     title = {Integral representations for multiple Hermite and multiple Laguerre polynomials},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {6},
     year = {2005},
     pages = {2001-2014},
     doi = {10.5802/aif.2148},
     mrnumber = {2187942},
     zbl = {1084.33008},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2005__55_6_2001_0}
}

M. BLEHER, Pavel; B.J. Kuijlaars, Arno. Integral representations for multiple Hermite and multiple Laguerre polynomials. Annales de l'Institut Fourier, Volume 55 (2005) no. 6, pp. 2001-2014. doi : 10.5802/aif.2148. https://aif.centre-mersenne.org/item/AIF_2005__55_6_2001_0/

[1] A. I. Aptekarev Multiple orthogonal polynomials, J. Comput. Appl. Math., Tome 99 (1998), pp. 423-447 | Article | MR 1662713 | Zbl 0958.42015

[2] A. I. Aptekarev; P. M. Bleher; A. B. J. Kuijlaars Large $n$ limit of Gaussian random matrices with external source, part II (to appear in Comm. Math. Phys., preprint math-ph/0408041, http://arxiv.org/abs/math-ph/0408041)

[3] A. I. Aptekarev; A. Branquinho; W. Van Assche Multiple orthogonal polynomials for classical weights, Trans. Amer. Math. Soc., Tome 355 (2003), pp. 3887-3914 | Article | MR 1990569 | Zbl 1033.33002

[4] J. Baik; G. Ben Arous; S. Péché Phase transition of the largest eigenvalue for non-null complex sample covariance matrices (to appear in Ann. Prob., preprint math.PR/0403022, http://arxiv.org/abs/math.PR/0403022)

[5] P. M. Bleher; A. B. J. Kuijlaars Random matrices with external source and multiple orthogonal polynomials, Internat. Math. Research Notices (2004), pp. 109-129 | MR 2038771 | Zbl 1082.15035

[6] P. M. Bleher; A. B. J. Kuijlaars Large $n$ limit of Gaussian random matrices with external source, part I, Commun. Math. Phys., Tome 252 (2004), pp. 43-76 | Article | MR 2103904 | Zbl 05071185

[7] A. Borodin Biorthogonal ensembles, Nuclear Phys., B, Tome 536 (1999), pp. 704-732 | MR 1663328 | Zbl 0948.82018

[8] E. Brézin; S. Hikami Correlations of nearby levels induced by a random potential, Nucl. Phys., B, Tome 479 (1996), pp. 697-706 | Article | MR 1418841 | Zbl 0925.82117

[9] E. Brézin; S. Hikami Spectral form factor in a random matrix theory, Phys. Rev., Tome E 55 (1997), pp. 4067-4083 | MR 1449379

[10] E. Brézin; S. Hikami Extension of level-spacing universality, Phys. Rev., Tome E 56 (1997), pp. 264-269

[11] E. Brézin; S. Hikami Universal singularity at the closure of a gap in a random matrix theory, Phys. Rev., Tome E 57 (1998), pp. 4140-4149 | MR 1618958

[12] E. Brézin; S. Hikami Level spacing of random matrices in an external source, Phys. Rev., Tome E 58 (1998), pp. 7176-7185 | MR 1662382

[13] E. Daems; A. B. J. Kuijlaars A Christoffel-Darboux formula for multiple orthogonal polynomials, J. Approx. Theory, Tome 130 (2004), pp. 188-200 | MR 2100703 | Zbl 1063.42014

[14] Harish-Chandra Differential operators on a semisimple Lie algebra, Amer. J. Math., Tome 79 (1957), pp. 87-120 | Article | MR 84104 | Zbl 0072.01901

[15] T. Imamura; T. Sasamoto Polynuclear growth model GOE${}^2$ and random matrix with deterministic source (preprint math-ph/0411057, http://arxiv.org/abs/math-ph/0411057)

[16] C. Itzykson; J. B. Zuber The planar approximation II, J. Math. Phys., Tome 21 (1980), pp. 411-421 | Article | MR 562985 | Zbl 0997.81549

[17] K. Johansson Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices, Comm. Math. Phys., Tome 215 (2001) no. 3, pp. 683-705 | Article | MR 1810949 | Zbl 0978.15020

[18] E. Nikishin; V. Sorokin Rational Approximation and Orthogonality, Translations of Mathematical Monographs, Tome 92, Amer. Math. Soc., Providence R.I, 1991 | MR 1130396 | Zbl 0733.41001

[19] C. Tracy; H. Widom The Pearcey process (preprint math.PR/0412005, http://arxiv.org/abs/math.PR/0412005) | Zbl 05118615

[20] W. Van Assche; E. Coussement Some classical multiple orthogonal polynomials, J. Comput. Appl. Math., Tome 127 (2001), pp. 317-347 | Article | MR 1808581 | Zbl 0969.33005

[21] P. Zinn-Justin Random Hermitian matrices in an external field, Nuclear Phys., Tome B 497 (1997), pp. 725-732 | MR 1463645 | Zbl 0933.82022

[22] P. Zinn-Justin Universality of correlation functions of Hermitian random matrices in an external field, Comm. Math. Phys, Tome 194 (1998), pp. 631-650 | Article | MR 1631489 | Zbl 0912.15028