I am aware of Hermite-Gaussian quadrature techniques for integrals of the form $$ \int_0^{\infty} f(x) \exp(-x^2) dx $$ However, am I looking for references on quadrature where the exponent is more generally $a \in (0, 2]$.
My understanding is that Hermite polynomials are applied in the special case above because they make for an orthogonal system with respect to $\exp(-x^2)$. I would therefore assume that a different kind of polynomials must be used when $a \neq 2$, but it is unclear to me exactly how one would find such polynomials.