Two variable Freud orthogonal polynomials and matrix Painleve-type difference equations

Abstract

We study bivariate orthogonal polynomials associated with Freud weight functions depending on real parameters. We analyse relations between the matrix coefficients of the three term relations for the orthonormal polynomials as well as the coefficients of the structure relations satisfied by these bivariate semiclassical orthogonal polynomials, also a matrix differential-difference equation for the bivariate orthogonal polynomials is deduced. The extension of the Painleve equation for the coefficients of the three term relations to the bivariate case and a two dimensional version of the Langmuir lattice are obtained.