Nonsymmetric linear difference equations for multiple orthogonal polynomials

We first give a brief survey of some aspects of orthogonal polynomials. The three-term recurrence relation gives a tridiagonal matrix and the corresponding Jacobi 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 polynomials are defined using orthogonality conditions spread out over r different 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 nonsymmetric. 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 finite order linear recurrence relation. In this section we will first recall some relevant facts from orthogonal polynomials (see Szegő [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 μ be a positive measure on the real line for which all the moments exist and for which the support contains infinitely many points. Without loss of generality we will normalize μ so that it is a probability measure. The monic orthogonal polynomials Pn (n = 0, 1, 2, . . .) for the measure μ are such that Pn(x) = x n + · · · has degree n and ∫ Pn(x)x k dμ(x) = 0, k = 0, 1, 2, . . . , n − 1. (1.1) ∗Research Director of the Belgian National Fund for Scientific Research (FWO). This research is supported by FWO research project G.0278.97 and INTAS 93-219ext.