Orthogonal polynomial projection kernel ($K_m(x,y)=\sqrt{w(x)w(y)}{\sum }_{j=0}^{m-1}{p}_j(x)p_j(y)/h_j$)

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 $1/m!$.

For $w(x)=x^a{e}^{-x}$ on $[0,\infty )$, $a>-1$, the monic Generalized Laguerre polynomials have squared norms $h_j=j!\Gamma (j+a+1)$. The corresponding orthogonal polynomial projection kernel is $(xy{)}^{a/2}{e}^{-(x+y)/2}{\sum }_{j=0}^{m-1}j!L_j^{(a)}(x)L_j^{(a)}(y)/\Gamma (j+a+1)$ for nonnegative $x,y$, and zero otherwise.