Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 7 (2011), 098, 31 pages      arXiv:1106.1307      https://doi.org/10.3842/SIGMA.2011.098

Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization

F. Alberto Grünbaum a, Manuel D. de la Iglesia b and Andrei Martínez-Finkelshtein c
a) Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720 USA
b) Departamento de Análisis Matemático, Universidad de Sevilla, Apdo (P.O. BOX) 1160, 41080 Sevilla, Spain
c) Departamento de Estadística y Matemática Aplicada, Universidad de Almería, 04120 Almería, Spain

Received June 09, 2011, in final form October 20, 2011; Published online October 25, 2011

Abstract
We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators will lead to a family of second-order differential equations satisfied by the orthogonal polynomials, some of them of 0-th and first order, something also impossible in the scalar setting. This shows that the differential properties in the matrix case are much more complicated than in the scalar situation. We will study several examples given in the last years as well as others not considered so far.

Key words: matrix orthogonal polynomials; Riemann-Hilbert problems.

pdf (580 Kb)   tex (35 Kb)

References

  1. Álvarez-Fernández C., Fidalgo U., Mañas M., The multicomponent 2D Toda hierarchy: generalized matrix orthogonal polynomials, multiple orthogonal polynomials and Riemann-Hilbert problems, Inverse Problems 26 (2010), 055009, 15 pages, arXiv:0911.0941.
  2. Berezans'ki Ju.M., Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968.
  3. Borrego J., Castro M.M., Durán A.J., Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits, arXiv:1102.1578.
  4. Cassatella-Contra G.A., Mañas M., Riemann-Hilbert problems, matrix orthogonal polynomials and discrete matrix equations with singularity confinement, Stud. Appl. Math., to appear, arXiv:1106.0036.
  5. Castro M.M., Grünbaum F.A., Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples, J. Nonlinear Math. Phys. 12 (2005), suppl. 2, 63-76.
  6. Castro M.M., Grünbaum F.A., The non-commutative bispectral problem for operators of order one, Constr. Approx. 27 (2008), 329-347.
  7. Chen Y., Ismail M.E.H., Ladder operators and differential equations for orthogonal polynomials, J. Phys. A: Math. Gen. 30 (1997), 7817-7829.
  8. Daems E., Kuijlaars A.B.J., Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions, J. Approx. Theory 146 (2007), 91-114, math.CA/0511470.
  9. Daems E., Kuijlaars A.B.J., Veys W., Asymptotics of non-intersecting Brownian motions and a 4×4 Riemann-Hilbert problem, J. Approx. Theory 153 (2008), 225-256, math.CV/0701923.
  10. Dai D., Kuijlaars A.B.J., Painlevé IV asymptotics for orthogonal polynomials with respect to a modified Laguerre weight, Stud. Appl. Math. 122 (2009), 29-83, arXiv:0804.2564.
  11. Damanik D., Pushnitski A., Simon B., The analytic theory of matrix orthogonal polynomials, Surv. Approx. Theory 4 (2008), 1-85, arXiv:0711.2703.
  12. Deift P.A., Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics, Vol. 3, New York University Courant Institute of Mathematical Sciences, New York, 1999.
  13. Deift P.A., Riemann-Hilbert methods in the theory of orthogonal polynomials, in Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon's 60th Birthday, Proc. Sympos. Pure Math., Vol. 76, Amer. Math. Soc., Providence, RI, 2007, 715-740, math.CA/0603309.
  14. Deift P.A., Gioev D., Random matrix theory: invariant ensembles and universality, Courant Lecture Notes in Mathematics, Vol. 18, Courant Institute of Mathematical Sciences, New York, 2009.
  15. Deift P.A., Kamvissis S., Kriecherbauer Th., Zhou X., The Toda rarefaction problem, Comm. Pure Appl. Math. 49 (1996), 35-83.
  16. Deift P.A., Zhou X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2) 137 (1993), 295-368, math.AP/9201261.
  17. Deift P.A., Zhou X., Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space, Comm. Pure Appl. Math. 56 (2003), 1029-1077, math.AP/0206222.
  18. Delvaux S., Average characteristic polynomials for multiple orthogonal polynomial ensembles, J. Approx. Theory 162 (2010), 1033-1067, arXiv:0907.0156.
  19. Delvaux S., Kuijlaars A.B.J., A phase transition for nonintersecting Brownian motions, and the Painlevé equation, Int. Math. Res. Not. 2009 (2009), no. 19, 3639-3725, arXiv:0809.1000.
  20. Durán A.J., Markov's theorem for orthogonal matrix polynomials, Canad. J. Math. 48 (1996), 1180-1195.
  21. Durán A.J., Matrix inner product having a matrix symmetric second order differential operator, Rocky Mountain J. Math. 27 (1997), 585-600.
  22. Durán A.J., Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices, J. Approx. Theory 161 (2009), 88-113.
  23. Durán A.J., Grünbaum F.A., Orthogonal matrix polynomials satisfying second-order differential equations, Int. Math. Res. Not. 2004 (2004), no. 10, 461-484.
  24. Durán A.J., Grünbaum F.A., A characterization for a class of weight matrices with orthogonal matrix polynomials satisfying second-order differential equations, Int. Math. Res. Not. 2005 (2005), no. 23, 1371-1390.
  25. Durán A.J., Ismail M.E.H., Differential coefficients of orthogonal matrix polynomials, J. Comput. Appl. Math. 190 (2006), 424-436.
  26. Durán A.J., Van Assche W., Orthogonal matrix polynomials and higher-order recurrence relations, Linear Algebra Appl. 219 (1995), 261-280, math.CA/9310220.
  27. Flaschka H., The Toda lattice. II. Existence of integrals, Phys. Rev. B 9 (1974), 1924-1925.
  28. Fokas A.S., Its A.R., Kitaev A.V., The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys. 147 (1992), 395-430.
  29. Grünbaum F.A., Matrix valued Jacobi polynomials, Bull. Sci. Math. 127 (2003), 207-214.
  30. Grünbaum F.A., Random walks and orthogonal polynomials: some challenges, in Probability, Geometry and Integrable Systems, Math. Sci. Res. Inst. Publ., Vol. 55, Cambridge Univ. Press, Cambridge, 2008, 241-260.
  31. Grünbaum F.A., Block tridiagonal matrices and a beefed-up version of the Ehrenfest urn model, in Modern Analysis and Applications. The Mark Krein Centenary Conference, Vol. 1, Operator Theory and Related Topics, Oper. Theory Adv. Appl., Vol. 100, Birkhäuser Verlag, Basel, 2009, 267-277.
  32. Grünbaum F.A., A spectral weight matrix for a discrete version of Walsh's spider, in Topics in Operator Theory, Vol. 1, Operators, Matrices and Analytic Functions, Oper. Theory Adv. Appl., Vol. 202, Birkhäuser Verlag, Basel, 2010, 253-264.
  33. Grünbaum F.A., de la Iglesia M.D., Matrix valued orthogonal polynomials arising from group representation theory and a family of quasi-birth-and-death processes, SIAM J. Matrix Anal. Appl. 30 (2008), 741-761.
  34. Grünbaum F.A., Pacharoni I., Tirao J., Matrix valued spherical functions associated to the complex projective plane, J. Funct. Anal. 188 (2002), 350-441, math.RT/0108042.
  35. Grünbaum F.A., Pacharoni I., Tirao J., Matrix valued orthogonal polynomials of the Jacobi type, Indag. Math. (N.S.) 14 (2003), 353-366.
  36. Grünbaum F.A., Pacharoni I., Tirao J., An invitation to matrix-valued spherical functions: linearization of products in the case of complex projective space P2(C), in Modern Signal Processing, Math. Sci. Res. Inst. Publ., Vol. 46, Cambridge Univ. Press, Cambridge, 2004, 147-160, math.RT/0202304.
  37. Grünbaum F.A., Pacharoni I., Tirao J., Two stochastic models of a random walk in the U(n) spherical duals of U(n+1), arXiv:1010.0720.
  38. Grünbaum F.A., Tirao J., The algebra of differential operators associate to a weight matrix, Integr. Equ. Oper. Theory 58 (2007), 449-475.
  39. Hille E., Ordinary differential equations in the complex domain, Pure and Applied Mathematics, Wiley-Interscience, New York, 1976.
  40. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
  41. Its A.R., Kuijlaars A.B.J., Östensson J., Asymptotics for a special solution of the thirty fourth Painlevé equation, Nonlinearity 22 (2009), 1523-1558, arXiv:0811.3847.
  42. Kren M.G., Infinite J-matrices and a matrix-moment problem, Doklady Akad. Nauk SSSR (N.S.) 69 (1949), 125-128 (in Russian).
  43. Kren M.G., Fundamental aspects of the representation theory of Hermitian operators with deficiency index (m,m), Am. Math. Soc. Translat. Ser. 2 97 (1971), 75-143.
  44. Kuijlaars A.B.J., Martínez-Finkelshtein A., Wielonsky F., Non-intersecting squared Bessel paths and multiple orthogonal polynomials for modified Bessel weights, Comm. Math. Phys. 286 (2009), 217-275, arXiv:1011.1278.
  45. Kuijlaars A.B.J., Multiple orthogonal polynomial ensembles, in Recent Trends in Orthogonal Polynomials and Approximation Theory, Contemp. Math., Vol. 507, Amer. Math. Soc., Providence, RI, 2010, 155-176, arXiv:0902.1058.
  46. McLaughlin K.T.-R., Vartanian A.H., Zhou X., Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights, IMRP Int. Math. Res. Pap. 2006 (2006), Art. ID 62815, 216 pages, math.CA/0601306.
  47. McLaughlin K.T.-R., Vartanian A.H., Zhou X., Asymptotics of Laurent polynomials of odd degree orthogonal with respect to varying exponential weights, Constr. Approx. 27 (2008), 149-202, math.CA/0601595.
  48. 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.
  49. Sibuya Y., Linear differential equations in the complex domain: problems of analytic continuation, Translations of Mathematical Monographs, Vol. 82, American Mathematical Society, Providence, RI, 1990.
  50. Simon B., Orthogonal polynomials on the unit circle I and II, AMS Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI, 2005.
  51. Simon B., Szegö's theorem and its descendants: spectral theory for L2 perturbations of orthogonal polynomials, Princeton University Press, 2010.
  52. Van Assche W., Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials, in Difference Equations, Special Functions and Orthogonal Polynomials, World Sci. Publ., Hackensack, NJ, 2007, 687-725, math.CA/0512358.


Previous article   Next article   Contents of Volume 7 (2011)