Darboux transforms on band matrices, weights, and associated polynomials.

Classically, it is well known that a single weight on a real interval leads to orthogonal polynomials. In Generalized orthogonal polynomials, discrete KP and Riemann-Hilbertproblems, Comm. Math. Phys. 207 (1999), 589–620, we have shown that m-periodic sequences of weights lead to “moments,” polynomials dened by determinants of matrices involving these moments and 2m + 1-step relations between them, thus leading to 2m + 1-band matrices L. Given a Darboux transformations on L, which effect does it have on the m-periodic sequence of weights and on the associated polynomials? These questions will receive a precise answer in this paper. The methods are based on introducing time parameters in the weights, making the band matrix L evolve according to the so-called discrete KP hierarchy. Darboux transformations on that L translate into vertex operators acting on the τ-function. [ABSTRACT FROM PUBLISHER]