Solved: The Laguerre polynomials {L0(x), L1(x)...} form an orthogonal set on [0,) and
Chapter 4, Problem 6(choose chapter or problem)
The Laguerre polynomials {L0(x), L1(x)...} form an orthogonal set on [0,) and satisfy 0 exLi(x)Lj(x) dx = 0, for i = j. (See Section 8.2.) The polynomial Ln(x) has n distinct zeros x1, x2, ... , xn in [0,). Let cn,i = 0 ex n j=1 j=i x xj xi xj dx. Show that the quadrature formula 0 f (x)ex dx = n i=1 cn,if (xi) has degree of precision 2n 1. (Hint: Follow the steps in the proof of Theorem 4.7.) 7
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