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For real monic orthogonal polynomials $p_j$ with squared weighted norms $h_j$ under a positive weight $w$, the functions $p_j\sqrt{w/h_j}$ form an orthonormal set for the unweighted reference measure. Their finite-rank projection kernel turns the squared weighted Vandermonde determinant into a kernel determinant. Normalization on the full labelled configuration space is $1/m!$.

Write ${\tilde{P}}_j(x)=x^{j-1}+{\sum }_{r=0}^{j-2}{b}_{rj}{x}^r$. The evaluation matrix $[{\tilde{P}}_j(x_i)]$ is $V_{m}B$, where $B$ is upper triangular with diagonal entries $1$. Its determinant is therefore unchanged. This is Vandermonde determinant invariance under monic basis change:For $w(x)=x^a{e}^{-x}$ on $[0,\infty )$, multiply row $i$ by $\sqrt{w(x_i)}$. The requested weighted product becomes the weighted Vandermonde determinant squareThe sign difference between the two orientations of the Vandermonde product disappears on squaring. For $a=0$ interpret the weight at zero by continuity, so $w(0)=1$; for $a>0$, $w(0)=0$.

Use the symmetric joint probability density function of exchangeably labelled eigenvalues, normalized over all of ${\mathbb{R}}^3$. The finite-rank projection kernel hasSuccessive applications of Gaudin's lemma therefore giveIt follows thatAs a normalization check, the weighted Vandermonde determinant yieldsIf instead one uses ordered eigenvalues on ${\lambda }_1<{\lambda }_2<{\lambda }_3$, the normalized density on that chamber is six times the labelled density. In that convention the determinant prefactor on the chamber is one.