Gauss quadrature rule with a specific weighting function

Question

I am wondering if anyone can point me to a Gauss quadrature rule on $[0,\infty)$ with $w(x)=x^2\ \mathrm{exp}(-x^2)$. The most similar thing that I can find is the one that is based on the generalized Laguerre polynomial with a weighting function $w(x)=x^a\ \mathrm{exp}(-x)$ here. Thanks!

Answer 1

Shizgal, B. (1981). A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems. Journal of Computational Physics 41, 309–328 doi:10.1016/0021-9991(81)90099-1

A new Gaussian quadrature procedure is developed for integrals of the form $\int_0^\infty \, e^{-y^2} y^p F(y) \, dy $ for $p$ = $0$, $1$ and $2$. Recursion relations are derived for the coefficients in the general three term recurrence relation for the polynomials whose roots are the quadrature abscissae. A comparison with the Gauss-Laguerre quadrature procedure is presented. Solutions of the chemical kinetic Boltzmann equation are obtained with a discrete ordinate method based on this Gaussian quadrature procedure. The results are compared with previous solutions obtained with a polynomial expansion method.