J. Coussement

Some discrete multiple orthogonal polynomials

In this paper, we extend the theory of discrete orthogonal polynomials (on a linear lattice) to polynomials satisfying orthogonality conditions with respect to r positive discrete measures. First we recall the known results of the classical orthogonal polynomials of Charlier, Meixner, Kravchuk and Hahn (T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978; R. Koekoek and R.F. Swarttouw, Reports of the Faculty of Technical Mathematics and Informatics No. 98-17, Delft, 1998; A.F. Nikiforov et al., Classical Orthogonal Polynomials of a Discrete Variable, Springer, Berlin, 1991). These polynomials have a lowering and raising operator, which give rise to a Rodrigues formula, a second order difference equation, and an explicit expression from which the coefficients of the three-term recurrence relation can be obtained. Then we consider r positive discrete measures and define two types of multiple orthogonal polynomials. The continuous case (Jacobi, Laguerre, Hermite, etc.) was studied by Van Assche and Coussement (J. Comput. Appl. Math. 127 (2001) 317-347) and Aptekarev et al. (Multiple orthogonal polynomials for classical weights, manuscript). The families of multiple orthogonal polynomials (of type II) that we will study have a raising operator and hence a Rodrigues formula. This will give us an explicit formula for the polynomials. Finally, there also exists a recurrence relation of order r + 1 for these multiple orthogonal polynomials of type II. We compute the coefficients of the recurrence relation explicitly when r = 2.

Asymptotic zero distribution for a class of multiple orthogonal polynomials

We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples of multiple orthogonal polynomials, in particular Jacobi-Pineiro, Laguerre I and the example associated with modified Bessel functions. We also discuss an application to Toeplitz matrices.

Multiple Wilson and Jacobi--Piòeiro polynomials

We introduce multiple Wilson polynomials, which give a new example of multiple orthogonal polynomials (Hermite-Pade polynomials) of type II. These polynomials can be written as a Jacobi-Pineiro transform, which is a generalization of the Jacobi transform for Wilson polynomials, found by Koornwinder. Here we need to introduce Jacobi and Jacobi-Pineiro polynomials with complex parameters. Some explicit formulas are provided for both Jacobi-Pineiro and multiple Wilson polynomials, one of them in terms of Kampe de Feriet series. Finally, we look at some limiting relations and construct a part of a multiple AT-Askey table.

Direct and inverse spectral transform for the relativistic Toda lattice and the connection with Laurent orthogonal polynomials

We introduce a spectral transform for the finite relativistic Toda lattice (RTL) in generalized form. In the nonrelativistic case, Moser constructed a spectral transform from the spectral theory of symmetric Jacobi matrices. Here we use a non-symmetric generalized eigenvalue problem for a pair of bidiagonal matrices (L, M) to define the spectral transform for the RTL. The inverse spectral transform is described in terms of a terminating T-fraction. The generalized eigenvalues are constants of motion and the auxiliary spectral data have explicit time evolution. Using the connection with the theory of Laurent orthogonal polynomials, we study the long-time behaviour of the RTL. As in the case of the Toda lattice the matrix entries have asymptotic limits. We show that L tends to an upper Hessenberg matrix with the generalized eigenvalues sorted on the diagonal, while M tends to the identity matrix.

Differential equations for multiple orthogonal polynomials with respect to classical weights: raising and lowering operators

We obtain a lowering operator for multiple orthogonal polynomials having orthogonality conditions with respect to r ∈ N classical weights. These multiple orthogonal polynomials are generalizations of the classical orthogonal polynomials. Combining the lowering operator with the raising operators, which have been obtained earlier in the literature, we then also obtain a linear differential equation of order r + 1.

An extension of the Toda lattice: a direct and inverse spectral transform connected with orthogonal rational functions

We introduce an extension of the finite Toda lattice, depending on a choice of time independent parameters , which also includes the finite relativistic Toda lattice as a limit case and comes from the theory of orthogonal rational functions. Generalizing the method Moser used in the case of the finite Toda lattice, we solve this system of nonlinear differential equations for some initial data with the aid of a spectral transform. In particular we study a generalized eigenvalue problem of a pair of matrices (J,I+JD) where J is a symmetric Jacobi matrix and D = diag (α0−1,α1−1,...,αN−1−1). The inverse spectral transform is described in terms of terminating continued fractions. Finally we compute the time evolution of the spectral data. Using the connection of this spectral transform with the theory of orthogonal rational functions we prove that the matrix J tends to a diagonal matrix containing the generalized eigenvalues and the parameters α0,α1,...,αN−1, thereby establishing the explicit long-time behaviour of the matrix entries.