  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.
