Riemann-Hilbert Problems for Multiple Orthogonal Polynomials

In the early nineties, Fokas, Its and Kitaev observed that there is a natural Riemann-Hilbert problem (for 2 x×2 matrix functions) associated with a system of orthogonal polynomials. This Riemann-Hilbert problem was later used by Deift et al. and Bleher and Its to obtain interesting results on orthogonal polynomials, in particular strong asymptotics which hold uniformly in the complex plane. In this paper we will show that a similar Riemann-Hilbert problem (for (r + 1) × (r + 1) matrix functions) is associated with multiple orthogonal polynomials. We show how this helps in understanding the relation between two types of multiple orthogonal polynomials and the higher order recurrence relations for these polynomials. Finally we indicate how an extremal problem for vector potentials is important for the normalization of the Riemann-Hilbert problem. This extremal problem also describes the zero behavior of the multiple orthogonal polynomials.

[1]  Андрей Александрович Гончар,et al.  Об аппроксимациях Эрмита - Паде для систем функций марковского типа@@@Hermite - Pade approximants for systems of Markov-type functions , 1997 .

[2]  Athanassios S. Fokas,et al.  Discrete Painlevé equations and their appearance in quantum gravity , 1991 .

[3]  P. Deift.  Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach , 2000 .

[4]  K. Driver,et al.  Normality in Nikishin systems , 1994 .

[5]  V. N. Sorokin,et al.  Rational Approximations and Orthogonality , 1991 .

[6]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1993 .

[7]  E. Saff,et al.  Logarithmic Potentials with External Fields , 1997 .

[8]  A. Ronveaux,et al.  On a system of “classical” polynomials of simultaneous orthogonality , 1996 .

[9]  E. Rakhmanov,et al.  On the equilibrium problem for vector potentials , 1985 .

[10]  A. Aptekarev,et al.  Multiple orthogonal polynomials , 1998 .

[11]  J Nuttall,et al.  Asymptotics of diagonal Hermite-Padé polynomials , 1984 .

[12]  Stephanos Venakides,et al.  Strong asymptotics of orthogonal polynomials with respect to exponential weights , 1999 .

[13]  V. A. Kaljagin.  ON A CLASS OF POLYNOMIALS DEFINED BY TWO ORTHOGONALITY RELATIONS , 1981 .

[14]  E. Rakhmanov,et al.  Hermite-Pade approximants for systems of Markov-type functions , 1997 .

[15]  H. Padé,et al.  Mémoire sur les développements en fractions continues de la fonction exponentielle, pouvant servir d'introduction à la théorie des fractions continues algébriques , 1899 .

[16]  A. Aptekarev.  Strong asymptotics of multiply orthogonal polynomials for Nikishin systems , 1999 .

[17]  Pavel Bleher,et al.  Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model , 1999, math-ph/9907025.

[18]  Athanassios S. Fokas,et al.  The isomonodromy approach to matric models in 2D quantum gravity , 1992 .

[19]  Athanassios S. Fokas,et al.  COMMUNICATIONS OF THE MOSCOW MATHEMATICAL SOCIETY: The isomonodromy approach in the theory of two-dimensional quantum gravitation , 1990 .

[20]  Stephanos Venakides,et al.  UNIFORM ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO VARYING EXPONENTIAL WEIGHTS AND APPLICATIONS TO UNIVERSALITY QUESTIONS IN RANDOM MATRIX THEORY , 1999 .

[21]  A. Aptekarev.  ASYMPTOTICS OF SIMULTANEOUSLY ORTHOGONAL POLYNOMIALS IN THE ANGELESCO CASE , 1989 .

[22]  Walter Van Assche,et al.  Multiple orthogonal polynomials, irrationality and transcendence , 1999 .

[23]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1992, math/9201261.