Gauss–Laguerre quadrature

Remarks

This is the Gaussian quadrature rule for the weight $e^{-x}$ on $[0, \infty)$, whose orthogonal polynomials are the $L_n$; the proof is the general one, by division by $L_n$ and orthogonality. Example: for $n = 2$, $L_2(x) = \frac12 (x^2 - 4x + 2)$ has zeros $2 \pm \sqrt{2}$ and $L_2'(x) = x - 2 = \pm\sqrt{2}$ there, so $w = \frac{1}{2(2 \pm \sqrt{2})} = \frac{2 \mp \sqrt{2}}{4}$. Check: the weights add up to $1 = \int_0^\infty e^{-x} dx$, and $\sum_i w_i x_i^2 = \frac{2 + \sqrt{2}}{2} + \frac{2 - \sqrt{2}}{2} = 2 = \int_0^\infty x^2 e^{-x} dx$. The degree $2n - 1$ cannot be raised: for $p = L_n^2$ the rule gives $0$ and the integral is positive. With the generalized Laguerre polynomials $L_n^{(\alpha)}$, $\alpha > -1$, the same construction gives rules for the weight $x^{\alpha} e^{-x}$.

Going deeper

For a student

To integrate $\int_0^\infty g(x)\,dx$, one writes $g(x) = e^{-x} \big(e^x g(x)\big)$ and applies the rule to $f(x) = e^x g(x)$. This works well when $f$ is smooth and grows at most like a polynomial. The rule is the half-line companion of Gauss–Hermite quadrature on the whole line and of Gauss–Legendre quadrature on a bounded interval.