G(t, x) = e-t+2tx -1 Hermite polynomials Hn (2) can be defined by the generating function H(a)t" n! n=0 (a) Use the gene

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G T X E T 2tx 1 Hermite Polynomials Hn 2 Can Be Defined By The Generating Function H A T N N 0 A Use The Gene 1 (135.64 KiB) Viewed 27 times

G(t, x) = e-t+2tx -1 Hermite polynomials Hn (2) can be defined by the generating function H(a)t" n! n=0 (a) Use the generating function to derive the relationships H(C) = 2nHn-1(x), Hn+1(x) = 2xHn(x) – 2nHn-1(2). (b) Use these relationships to derive a second-order ODE for which y = Hn (2) is a solution. Rewrite this equation in Sturm-Liouville form as d p(3) = -wany, - < x < 00, do dr for some p(x) and w(x) that you should specify, and hence deduce that Hn(x)Hm(x)e -r* dx = 0, dy . [ | =, L n2 for any m En (you may quote standard results for Sturm-Liouville theory from the lectures). (c) For a given integer value of n > 0, use your answers to (a) and (b) to determine conditions on the possible values of m that will ensure that H,(x)Hm (2)x²e-4° dx = 0). (d) Show that a=*. | G(t, x)G(3, 1)e =" dx = Vredst 2 2 and use this result to compute |H}(r)e ==* dr. [HINT: you may quote without proof that see-u? du = VT. 00 =