Orthogonal polynomials through the invariant theory of binary forms
Rings and Algebras
2014-10-20 v1
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.
Cite
@article{arxiv.1410.4683,
title = {Orthogonal polynomials through the invariant theory of binary forms},
author = {Pasquale Petrullo and Domenico Senato and Rosaria Simone},
journal= {arXiv preprint arXiv:1410.4683},
year = {2014}
}