Geodesic
The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Consider the Laguerre polynomials and deform them by the introduction in the measure of an exponential singularity at zero. In [Chen Y., Its A., J. Approx. Theory 162 (2010), 270–297] the authors proved that this deformation can be described by systems of differential/difference equations for the corresponding recursion coefficients and that these equations, ultimately, are equivalent to the Painlevé III equation and its Bäcklund/Schlesinger transformations. Here we prove that an analogue result holds for some kind of semiclassical matrix-valued orthogonal polynomials of Laguerre type.
Keywords: Painlevé equations; Toda lattices; Riemann–Hilbert problems; matrix-valued orthogonal polynomials. 
@article{SIGMA_2018_14_a75,
     author = {Mattia Cafasso and Manuel D. De La Iglesia},
     title = {The {Toda} and {Painlev\'e} {Systems} {Associated} with {Semiclassical} {Matrix-Valued} {Orthogonal} {Polynomials} of {Laguerre} {Type}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2018},
     volume = {14},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a75/}
}
TY  - JOUR
AU  - Mattia Cafasso
AU  - Manuel D. De La Iglesia
TI  - The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2018
VL  - 14
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a75/
LA  - en
ID  - SIGMA_2018_14_a75
ER  - 
%0 Journal Article
%A Mattia Cafasso
%A Manuel D. De La Iglesia
%T The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type
%J Symmetry, integrability and geometry: methods and applications
%D 2018
%V 14
%U http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a75/
%G en
%F SIGMA_2018_14_a75
Mattia Cafasso; Manuel D. De La Iglesia. The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a75/

[1] Adler M., van Moerbeke P., Vanhaecke P., “Moment matrices and multi-component KP, with applications to random matrix theory”, Comm. Math. Phys., 286 (2009), 1–38, arXiv: math-ph/0612064 | DOI | MR | Zbl

[2] Álvarez-Fernández C., Mañas M., “Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies”, Adv. Math., 240 (2013), 132–193, arXiv: 1202.2898 | DOI | MR | Zbl

[3] Ariznabarreta G., Mañas M., “Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems”, Adv. Math., 264 (2014), 396–463, arXiv: 1312.0150 | DOI | MR | Zbl

[4] Ariznabarreta G., Mañas M., “Multivariate orthogonal polynomials and integrable systems”, Adv. Math., 302 (2016), 628–739, arXiv: 1409.0570 | DOI | MR | Zbl

[5] Bertola M., Cafasso M., “Fredholm determinants and pole-free solutions to the noncommutative Painlevé II equation”, Comm. Math. Phys., 309 (2012), 793–833, arXiv: 1101.3997 | DOI | MR | Zbl

[6] Cafasso M., “Matrix biorthogonal polynomials on the unit circle and non-abelian Ablowitz–Ladik hierarchy”, J. Phys. A: Math. Theor., 42 (2009), 365211, 20 pp., arXiv: 0804.3572 | DOI | MR | Zbl

[7] Cafasso M., de la Iglesia M. D., “Non-commutative Painlevé equations and Hermite-type matrix orthogonal polynomials”, Comm. Math. Phys., 326 (2014), 559–583, arXiv: 1301.2116 | DOI | MR | Zbl

[8] Cassatella-Contra G. A., Mañas M., “Riemann–Hilbert problems, matrix orthogonal polynomials and discrete matrix equations with singularity confinement”, Stud. Appl. Math., 128 (2012), 252–274, arXiv: 1106.0036 | DOI | MR | Zbl

[9] Chen Y., Its A., “Painlevé III and a singular linear statistics in Hermitian random matrix ensembles. I”, J. Approx. Theory, 162 (2010), 270–297, arXiv: 0808.3590 | DOI | MR | Zbl

[10] Durán A.J., Grünbaum F. A., “Orthogonal matrix polynomials satisfying second-order differential equations”, Int. Math. Res. Not., 2004 (2004), 461–484 | DOI | Zbl

[11] Durán A.J., López-Rodríguez P., “Structural formulas for orthogonal matrix polynomials satisfying second-order differential equations. II”, Constr. Approx., 26 (2007), 29–47 | DOI | MR | Zbl

[12] Erdélyi A., “Über einige bestimmte Integrale, in denen die Whittakerschen $M^{k,m}$-Funktionen auftreten”, Math. Z., 40 (1936), 693–702 | DOI | MR

[13] Etingof P., Gelfand I., Retakh V., “Nonabelian integrable systems, quasideterminants, and Marchenko lemma”, Math. Res. Lett., 5 (1998), 1–12, arXiv: q-alg/9707017 | DOI | MR | Zbl

[14] Fokas A. S., Its A. R., Kitaev A. V., “The isomonodromy approach to matrix models in $2$D quantum gravity”, Comm. Math. Phys., 147 (1992), 395–430 | DOI | MR | Zbl

[15] Grünbaum F. A., de la Iglesia M. D., Martínez-Finkelshtein A., “Properties of matrix orthogonal polynomials via their Riemann–Hilbert characterization”, SIGMA, 7 (2011), 098, 31 pp., arXiv: 1106.1307 | DOI | Zbl

[16] Johansson K., “Discrete orthogonal polynomial ensembles and the {P}lancherel measure”, Ann. of Math., 153 (2001), 259–296, arXiv: math.CO/9906120 | DOI | MR | Zbl

[17] Miranian L., “Matrix-valued orthogonal polynomials on the real line: some extensions of the classical theory”, J. Phys. A: Math. Gen., 38 (2005), 5731–5749 | DOI | MR | Zbl

[18] Moser J., “Finitely many mass points on the line under the influence of an exponential potential – an integrable system”, Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lecture Notes in Phys., 38, Springer, Berlin, 1975, 467–497 | DOI

[19] Retakh V., Rubtsov V., “Noncommutative Toda chains, Hankel quasideterminants and the Painlevé II equation”, J. Phys. A: Math. Theor., 43 (2010), 505204, 13 pp., arXiv: 1007.4168 | DOI | Zbl