History of Nodes and weights of Gauss–Laguerre quadrature

         equals: HREF{One}         number: '1'-        comment: $w_1=1=\int_0^\infty e^{-x}\,dx$+        comment: $w_1=1=\int_0^\infty e^{-x}\,\mathrm{d}x$       x:         equals: HREF{One}

     at a root of $L_n$. Also $\dfrac{1}{w_k}=\sum_{j=0}^{n-1}L_j(x_k)^2$, the Christoffel     function at the node, the $L_j$ being orthonormal for $e^{-x}$.-  formula-exactness: $\sum_{k=1}^{n} w_k x_k^m=\int_{0}^{\infty}x^m e^{-x}\,dx=m!$+  formula-exactness: $\sum_{k=1}^{n} w_k x_k^m=\int_{0}^{\infty}x^m e^{-x}\,\mathrm{d}x=m!$     for $0\leq m\leq 2n-1$, and not for $m=2n$; the moments are in HREF{Factorial}[the     table of factorials]. In particular $w_k>0$ and $\sum_{k=1}^{n}w_k=1$.-  formula-secondary: $w_k=\dfrac{q_n(x_k)}{L_n'(x_k)}$ with $q_n(x)=\int_{0}^{\infty}\dfrac{L_n(t)-L_n(x)}{t-x}\,e^{-t}\,dt$+  formula-secondary: $w_k=\dfrac{q_n(x_k)}{L_n'(x_k)}$ with $q_n(x)=\int_{0}^{\infty}\dfrac{L_n(t)-L_n(x)}{t-x}\,e^{-t}\,\mathrm{d}t$     the secondary polynomial of $L_n$ for the density $e^{-x}$, a polynomial of degree     $n-1$ with rational coefficients; the same construction for the HREF{Legendre_polynomials}[Legendre

     with $2n$ points and $u_k$ their weights; for $x^{1/2}e^{-x}$ it has nodes $t_k^2$     and weights $2u_k t_k^2$, taken from the Gauss–Hermite rule with $2n+1$ points.-    Both follow from $\int_{0}^{\infty}g(x)\,x^{\mp 1/2}e^{-x}\,dx=\int_{-\infty}^{\infty}-    g(t^2)\,t^{1\mp 1}\,e^{-t^2}\,dt$ and the uniqueness of the Gauss rule.-  formula-whole-half-line: $\int_{0}^{\infty}f(x)\,dx\approx\sum_{k=1}^{n}w_k\,e^{x_k}f(x_k)$-    CITE{Wiki}, and on $[a,\infty)$ with the weight $e^{-x}$, $\int_{a}^{\infty}f(x)\,e^{-x}\,dx\approx+    Both follow from $\int_{0}^{\infty}g(x)\,x^{\mp 1/2}e^{-x}\,\mathrm{d}x=\int_{-\infty}^{\infty}+    g(t^2)\,t^{1\mp 1}\,e^{-t^2}\,\mathrm{d}t$ and the uniqueness of the Gauss rule.+  formula-whole-half-line: $\int_{0}^{\infty}f(x)\,\mathrm{d}x\approx\sum_{k=1}^{n}w_k\,e^{x_k}f(x_k)$+    CITE{Wiki}, and on $[a,\infty)$ with the weight $e^{-x}$, $\int_{a}^{\infty}f(x)\,e^{-x}\,\mathrm{d}x\approx     e^{-a}\sum_{k=1}^{n}w_k\,f(x_k+a)$. Programs:

     language: Python Definition: For $n\geq 1$ the Gauss–Laguerre quadrature rule with $n$ points is the-  unique rule $\int_{0}^{\infty} f(x)\,e^{-x}\,dx \approx \sum_{k=1}^{n} w_k f(x_k)$-  that is exact for every polynomial $f$ of degree at most $2n-1$ CITE{Wiki}. Listed-  are its nodes $x_1<\cdots<x_n$ and its weights $w_k$.+  unique rule $\int_{0}^{\infty} f(x)\,e^{-x}\,\mathrm{d}x \approx \sum_{k=1}^{n}+  w_k f(x_k)$ that is exact for every polynomial $f$ of degree at most $2n-1$ CITE{Wiki}.+  Listed are its nodes $x_1<\cdots<x_n$ and its weights $w_k$. Parameters:   n:

     half-line it gives the rules for $x^{\pm 1/2}e^{-x}$ - table: HREF{Factorial}[Factorial of natural numbers]-  relation: the moments $\int_0^\infty x^m e^{-x}\,dx=m!$ that the rule reproduces+  relation: the moments $\int_0^\infty x^m e^{-x}\,\mathrm{d}x=m!$ that the rule reproduces     for $m\leq 2n-1$ - table: HREF{Algebraic_numbers_of_degree_2}[Algebraic numbers of degree 2]