An extension of the associated rational functions on the unit circle

A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Caratheodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients. The new class under study includes the associated rational functions as a particular case. As a consequence of the previous general analysis, we obtain explicit representations for the associated rational functions of arbitrary order, as well as for the related Caratheodory function. Such representations are used to find new properties of the associated rational functions.Comment: 27 page

Rational Solutions of the Painlev\'e-II Equation Revisited

The rational solutions of the Painlev\'e-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann-Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solutions in the limit of large degree (equivalently the large-parameter limit). We review the elementary properties of the rational Painlev\'e-II functions, and then we describe three different Riemann-Hilbert representations of them that have appeared in the literature: a representation by means of the isomonodromy theory of the Flaschka-Newell Lax pair, a second representation by means of the isomonodromy theory of the Jimbo-Miwa Lax pair, and a third representation found by Bertola and Bothner related to pseudo-orthogonal polynomials. We prove that the Flaschka-Newell and Bertola-Bothner Riemann-Hilbert representations of the rational Painlev\'e-II functions are explicitly connected to each other. Finally, we review recent results describing the asymptotic behavior of the rational Painlev\'e-II functions obtained from these Riemann-Hilbert representations by means of the steepest descent method

Dilations and constrained algebras

It is well known that unital contractive representations of the disk algebra are completely contractive. Let A denote the subalgebra of the disk algebra consisting of those functions f whose first derivative vanishes at 0. We prove that there are unital contractive representations of A which are not completely contractive, and furthermore provide a Kaiser and Varopoulos inspired example for A and present a characterization of those contractive representations of A which are completely contractive. In the positive direction, for the algebra of rational functions with poles off the distinguished variety V in the bidisk determined by (z-w)(z+w)=0, unital contractive representations are completely contractive.Comment: New to version 2 is a proof of rational dilation for the distinguished variety in the bidisk determined by (z-w)(z+w)=

The Atiyah--Hitchin bracket and the open Toda lattice

The dynamics of finite nonperiodic Toda lattice is an isospectral deformation of the finite three--diagonal Jacobi matrix. It is known since the work of Stieltjes that such matrices are in one--to--one correspondence with their Weyl functions. These are rational functions mapping the upper half--plane into itself. We consider representations of the Weyl functions as a quotient of two polynomials and exponential representation. We establish a connection between these representations and recently developed algebraic--geometrical approach to the inverse problem for Jacobi matrix. The space of rational functions has natural Poisson structure discovered by Atiyah and Hitchin. We show that an invariance of the AH structure under linear--fractional transformations leads to two systems of canonical coordinates and two families of commuting Hamiltonians. We establish a relation of one of these systems with Jacobi elliptic coordinates.Comment: 26 pages, 2 figure