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3. Hermite polynomials Hn (2) can be defined by the generating function Hn(x)t" Σ n! G(t, x) = e-tº+2tx n=0 = (a) Use th
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3. Hermite polynomials Hn (2) can be defined by the generating function Hn(x)t" Σ n! G(t, x) = e-tº+2tx n=0 = (a) Use th
3. Hermite polynomials Hn (2) can be defined by the generating function Hn(x)t" Σ n! G(t, x) = e-tº+2tx n=0 = (a) Use the generating function to derive the relationships H (2) = 2nHn-1(x), Hn+1(x) = 2x Hn (x) – 2nHn-1(2). (b) Use these relationships to derive a second-order ODE for which y = Hn (2) y is a solution. Rewrite this equation in Sturm-Liouville form as d dy - < x < 0, dx dx for some p(x) and w(x) that you should specify, and hence deduce that [play id) = -2(a)ny, ro [()Hm(z)e== * dx = 0, = o for any m #n (you may quote standard results for Sturm-Liouville theory from the lectures).
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