These orthogonal polynomials are defined by H_eo(1) = 1 and

As is true for many special functions, the literature contains more than one notation, and one sometimes defines as Hermite polynomials the functions

This differs from our definition, which is preferred in applications.

(a) A generating function of the Hermite polynomials is

because H_en(x) = n! α_n(x). Prove this. Use the formula for the coefficients of a Maclaurin series and note that tx - 1/2t² = 1/2x² - 1/2(x - t²).

(b) Differentiating the generating function with respect to x, show that

(c) Orthogonality on the x-Axis needs a weight function that goes to zero sufficiently fast as x †’ ±ˆž, (Why?)

Show that the Hermite polynomials are orthogonal on -ˆž < x < ˆž with respect to the weight function r(x) = e^-x2/2. Use integration by parts and (21).

(d) Show that

Using this with n - 1 instead of n and (21), show that y = H_en(x) satisfies the ODE

Show that w = e^-x2/4y is a solution of Weber€™s equation

Transcribed Image Text:

(19) He,(x) = (1)"2 d" (e n = 1, 2, -... drr H = 1, HRCr) = (-1)"d"e¯² %3D