  The Beauville-Fujiki relation for a compact Hyperk\"ahler manifold $X$ of
dimension $2k$ allows to equip the symmetric power $\text{Sym}^kH^2(X)$ with a
symmetric bilinear form induced by the Beauville-Bogomolov form. We study some
of its properties and compare it to the form given by the Poincar\'e pairing.
  The construction generalizes to a definition for an induced symmetric
bilinear form on the symmetric power of any free module equipped with a
symmetric bilinear form. We point out how the situation is related to the
theory of orthogonal polynomials in several variables. Finally, we construct a
basis of homogeneous polynomials that are orthogonal when integrated over the
unit sphere $\mathbb{S}^d$, or equivalently, over $\mathbb{R}^{d+1}$ with a
Gaussian kernel.

Algebraic Geometry;Classical Analysis and ODEs;Combinatorics