When suitably normalized, the polynomial solutions to Equation (11.3.13) are called the

Chapter 11, Problem 10

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When suitably normalized, the polynomial solutions to Equation (11.3.13) are called the Hermite polynomials, and are denoted by HN (x). (a) Use Equation (11.3.13) to show that HN (x) satisfies (ex2 H N ) + 2N ex2 HN = 0. (11.3.14) [Hint: Replace with N in Equation (11.3.13) and multiply the resulting equation by ex2 .] (b) Use Equation (11.3.14) to prove that the Hermite polynomials satisfy ex2 HN (x)HM (x)dx = 0, M = N. (11.3.15) [Hint: Follow the steps taken in proving orthogonality of the Legendre polynomials. You will need to recall that lim x ex2 p(x) = 0, for any polynomial p.] (c) Let p(x) be a polynomial of degree N. Then we can write p(x) = N k=1 ak Hk (x). (11.3.16) Given that ex2 H2 N (x)dx = 2N N! , use (11.3.15) to prove that the constants in (11.3.16) are given by a j = 1 2 j j! ex2 Hj(x)p(x)dx.