Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems

Abstract

Classically, a single weight on an interval of the real line leads to moments, orthogonal polynomials and tridiagonal matrices. Appropriately deforming this weight with times t=(t_1,t_2,...), leads to the standard Toda lattice and tau-functions, expressed as Hermitian matrix integrals. This paper is concerned with a sequence of t-perturbed weights, rather than one single weight. This sequence leads to moments, polynomials and a (fuller) matrix evolving according to the discrete KP-hierarchy. The associated tau-functions have integral, as well as vertex operator representations. Among the examples considered, we mention: nested Calogero-Moser systems, concatenated solitons and m-periodic sequences of weights. The latter lead to 2m+1-band matrices and generalized orthogonal polynomials, also arising in the context of a Riemann-Hilbert problem. We show the Riemann-Hilbert factorization is tantamount to the factorization of the moment matrix into the product of a lower- times upper-triangular matrix.