On Generating Gaussian Quadrature Rules

Given a mass distribution dσ(x) on the (finite or infinite) interval (a,b), where σ(x) has at least n+1 points of increase, and assuming the existence of the first 2n moments of dσ(x), $${\mu _k} = \int_a^b {{x^k}} d\sigma (x),\;\;\;\;k = 0,1,2,...,2n - 1$$ (1.1) it is well known that the n-point Gaussian quadrature rule associated with the distribution da(x) exists and is unique. That is, there exist unique nodes v (n) ∊ (a,b) and weights λ v (n) > 0 such that $$\int_a^b {f(x)} d\sigma (x) = \sum\limits_{v = 1}^n {\lambda _v^{(n)}f(\xi _v^{(n)})} + {R_n}(f)$$ (1.2) with $$ {R_n}(f) = 0\,for\,all\,f \in {P_{2n - 1}} $$ (1.3) .