Q3 (20 points) Hermite polynomials H.(x) can be defined by the generating function G(1.x) = 2x H.(x)" (a) Use the genera

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Q3 20 Points Hermite Polynomials H X Can Be Defined By The Generating Function G 1 X 2x H X A Use The Genera 1 (24.46 KiB) Viewed 16 times

Q3 (20 points) Hermite polynomials H.(x) can be defined by the generating function G(1.x) = 2x H.(x)" (a) Use the generating function to derive the relationships H.(x) = 2nH.-.(x). HA+1(x) = 2xH.(x) – 2nH.- (x). (b) Use these relationships to derive a second-order ODE for which y = H.(x) is a solution. Rewrite this equation in Sturm--Liouville form as as (powiat) --werny. --0<x<co. for some p(x) and w(x) that you should specify, and hence deduce that S*H,()}_{2}&=* dx = 0. for any mo n (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 L*H.()H()?e=* dx = 0, (d) Show that | GU, G, X. * 4x = and use this result to compute L*}{* dx. [HINT: you may quote without proof that du = V.)