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We present an algebraic theory of orthogonal polynomials in several variablesthat includes classical orthogonal polynomials as a special case. Our bottomline is a straightforward connection between apolarity of binary forms and theinner product provided by a linear functional defined on a polynomial ring.Explicit determinantal formulae and multivariable extension of the Heineintegral formula are stated. Moreover, a general family of covariants thatincludes transvectants is introduced. Such covariants turn out to be theaverage value of classical basis of symmetric polynomials over a set of rootsof suitable orthogonal polynomials.

Orthogonal polynomials through the invariant theory of binary forms