Nonsymmetric Linear Diierence Equations for Multiple Orthogonal Polynomials

We rst give a brief survey of some aspects of orthogonal polynomials. The three-term recurrence relation gives a tridiagonal matrix and the corresponding Ja-cobi operator gives useful information about the orthogonalizing measure and the asymptotic behavior of the zeros of the orthogonal polynomials. The Toda lattice and other similar dynamical systems (Langmuir lattice or Kac-Van Moerbeke lattice) can be solved explicitly using Jacobi operators. Then we present multiple orthogonal polynomials, which are less known. These multiple orthogonal polynomi-als are deened using orthogonality conditions spread out over r diierent measures. There is a higher order recurrence relation with r + 2 terms, which gives a banded Hessenberg matrix and a corresponding operator which is essentially nonsymmet-ric. We give some examples and indicate how one can start working out a spectral theory for such operators. As an application we show that one can explicitly solve the Bogoyavlenskii lattice using certain multiple orthogonal polynomials. 1 Orthogonal Polynomials In this paper we will introduce multiple orthogonal polynomials and show how they are related to certain nonsymmetric linear operators that correspond to a nite order linear recurrence relation. In this section we will rst recall some relevant facts from orthogonal polynomials (see Szeg} o 25] or 27] for a more thorough treatment) and in the next section we will see how some of these facts have an extension to multiple orthogonal polynomials, but that the new setting is richer and still needs further study (see Nikishin and Sorokin 20] and Aptekarev 2] for more information on multiple orthogonal polynomials). Let be a positive measure on the real line for which all the moments exist and for which the support contains innnitely many points. Without loss of generality we will normalize so that it is a probability measure. The monic orthogonal polynomials P n (n = 0; 1; 2; : : :) for the measure are such that P n (x) = x n + has degree n and