Full length article: Multiple Meixner-Pollaczek polynomials and the six-vertex model

Authors:
Martin Bender;Steven Delvaux;Arno B. J. Kuijlaars
Affiliations:
MSRI, 17 Gauss Way, Berkeley, CA 94720-5070, United States;Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium;Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium
Venue:
Journal of Approximation Theory
Year:
2011

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Abstract

We study multiple orthogonal polynomials of Meixner-Pollaczek type with respect to a symmetric system of two orthogonality measures. Our main result is that the limiting distribution of the zeros of these polynomials is one component of the solution to a constrained vector equilibrium problem. We also provide a Rodrigues formula and closed expressions for the recurrence coefficients. The proof of the main result follows from a connection with the eigenvalues of (locally) block Toeplitz matrices, for which we provide some general results of independent interest. The motivation for this paper is the study of a model in statistical mechanics, the so-called six-vertex model with domain wall boundary conditions, in a particular regime known as the free fermion line. We show how the multiple Meixner-Pollaczek polynomials arise in an inhomogeneous version of this model.