Chapter 4.9, Problem 6ES

The Laguerre polynomials {L₀(x), L₁(x) ...} form an orthogonal set on [0, ∞) and satisfy ${\displaystyle {\int }_0^{\infty }{e}^{-x}{L}_i(x)L_j(x)dx}\text{ = 0}$, for i ≠ j. (See Section 8.2.) The polynomial L_n(x) has n distinct zeros x₁, x₂, . . . , x_n in [0, ∞). Let

$C_{n,i}={\displaystyle {\int }_0^{\infty }{e}^{-x}{\displaystyle \underset{\begin{array}{l}j=1\\ j\ne i\end{array}}{\overset{n}{\prod }}\frac{x-x_j}{x_i-x_j}dx.}}$

Show that the quadrature formula

${\displaystyle {\int }_0^{\infty }f(x)e^{-x}dx={\displaystyle \underset{i=1}{\overset{n}{\sum }}{C}_{n,i}f(x_i)}}$

has degree of precision 2n − 1. [Hint: Follow the steps in the proof of Theorem 4.7.]