Orthogonal polynomials through the invariant theory of binary forms

TL;DR

This paper develops an algebraic framework linking invariant theory of binary forms to multivariable orthogonal polynomials, providing explicit formulas and general covariants that unify classical and multivariable cases.

Contribution

It introduces a novel algebraic approach connecting apolarity and inner products to construct multivariable orthogonal polynomials with explicit formulas.

Findings

Explicit determinantal formulae for multivariable orthogonal polynomials

Extension of Heine integral formula to multiple variables

Introduction of covariants as averages over roots of orthogonal polynomials

Abstract

We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner product provided by a linear functional defined on a polynomial ring. Explicit determinantal formulae and multivariable extension of the Heine integral formula are stated. Moreover, a general family of covariants that includes transvectants is introduced. Such covariants turn out to be the average value of classical basis of symmetric polynomials over a set of roots of suitable orthogonal polynomials.