History of Nodes and weights of Gauss–Legendre quadrature

 Title: Nodes and weights of Gauss–Legendre quadrature Definition: For $n\geq 1$ the Gauss–Legendre quadrature rule with $n$ points is the-  unique rule $\int_{-1}^{1} f(x)\,dx \approx \sum_{k=1}^{n} w_k f(x_k)$ that is exact-  for every polynomial $f$ of degree at most $2n-1$ CITE{Wiki}. Listed are its nodes-  $x_1<\cdots<x_n$, the roots of the Legendre polynomial $P_n$, and its weights $w_k$.+  unique rule $\int_{-1}^{1} f(x)\,\mathrm{d}x \approx \sum_{k=1}^{n} w_k f(x_k)$+  that is exact for every polynomial $f$ of degree at most $2n-1$ CITE{Wiki}. Listed+  are its nodes $x_1<\cdots<x_n$, the roots of the Legendre polynomial $P_n$, and+  its weights $w_k$. Parameters:   n:

   formula-weights: $w_k=\dfrac{2}{(1-x_k^2)\,P_n'(x_k)^2}=\dfrac{2\,(1-x_k^2)}{n^2\,P_{n-1}(x_k)^2}$     CITE{AS}.-  formula-exactness: $\sum_{k=1}^{n} w_k x_k^m=\int_{-1}^{1}x^m\,dx=\dfrac{1+(-1)^m}{m+1}$+  formula-exactness: $\sum_{k=1}^{n} w_k x_k^m=\int_{-1}^{1}x^m\,\mathrm{d}x=\dfrac{1+(-1)^m}{m+1}$     for $0\leq m\leq 2n-1$, and not for $m=2n$.   formula-symmetry: $x_{n+1-k}=-x_k$, $w_{n+1-k}=w_k$, $w_k>0$ and $\sum_{k=1}^{n}w_k=2$.   formula-central-weight: 'For odd $n$, $x_{(n+1)/2}=0$ and $w_{(n+1)/2}=\dfrac{2}{P_n''(0)^2}=\dfrac{2\,((n-1)!!)^2}{n^2\,((n-2)!!)^2}$:     $2$, $8/9$, $128/225$, $512/1225$, $32768/99225$ for $n=1,3,5,7,9$.'-  formula-lagrange: $w_k=\int_{-1}^{1}\ell_k(x)\,dx$ with $\ell_k(x)=\prod_{j\neq+  formula-lagrange: $w_k=\int_{-1}^{1}\ell_k(x)\,\mathrm{d}x$ with $\ell_k(x)=\prod_{j\neq     k}\frac{x-x_j}{x_k-x_j}$ the Lagrange basis polynomial of the nodes; the same     integrals over HREF{Lagrange_basis_polynomials_for_equally_spaced_nodes}[equally