On multivariate orthogonal polynomials and elementary symmetric functions

Abstract

We study families of multivariate orthogonal polynomials with respect to the symmetric weight function in d variables Bγ(x)=∏i=1dω(xi)∏i<j|xi-xj|2γ+1,x∈(a,b)d,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} B_{\\gamma }(\\mathtt {x}) = \\prod \\limits _{i=1}^{d} \\omega (x_{i}) \\prod \\limits _{i<j} |x_{i}-x_{j}|^{2\\gamma +1}, \\quad \\mathtt {x}\\in (a,b)^{d}, \\end{aligned}$$\\end{document}for gamma >-1, where omega (t) is an univariate weight function in t in (a,b) and mathtt {x} = (x_{1},x_{2}, ldots , x_{d}) with x_{i} in (a,b). Applying the change of variables x_{i},i=1,2,ldots ,d, into u_{r},r=1,2,ldots ,d, where u_{r} is the r-th elementary symmetric function, we obtain the domain region in terms of the discriminant of the polynomials having x_{i},i=1,2,ldots ,d, as its zeros and in terms of the corresponding Sturm sequence. Choosing the univariate weight function as the Hermite, Laguerre, and Jacobi weight functions, we obtain the representation in terms of the variables u_{r} for the partial differential operators such that the respective Hermite, Laguerre, and Jacobi generalized multivariate orthogonal polynomials are the eigenfunctions. Finally, we present explicitly the partial differential operators for Hermite, Laguerre, and Jacobi generalized polynomials, for d=2 and d=3 variables.