I need some help if possible with the following problems.The statements are long so please bare with me.

This is concerned with a system of polynomials, the Hermite polynomials,

which arise in various applications, in particular in the analysis of the normal distri-

bution in probability and in the solution of certain dierential equations from physics.

The project explores the properties of these polynomials, including the dierential

equation they satisfy.The polynomials w are in fact a variant of the standard

Hermite polynomials. This version is more convenient for the applications to

probability, and differs from the usual version by a sign and a scaling of the x-variable.

a)The polynomials are defined as follows H_n(x) for n = 0; 1; : : : as follows: first, set H_0(x) =1 and H_1(x) = -x; then, for n 2, H_n is defined by the recurrence

H_n(x) = -xH_(n-1)(x) - (n - 1)H_(n-2)(x): (1)

I have to use (1) to verify that H_2(x) = x^2 -1 and H_3(x) = 3x-x^3, and calculate H_4(x)and H_5(x).

I also have to show that that H_n is an even function when n is even, and that it is an odd function when n is odd (maybe using (1) and induction on n).

Also that also (maybe I should still use induction and (1)) that

H_2k(0) = (-1)^k(2k -1)(2k -3) ....1:

What is the value of Hn(0) when n is odd?

b)Here I must show that t H_n satises a dierential equation. By dierentiating (1) and

using induction on n, show that, for n >= 1,

H'_n(x) = -nH_(n-1)(x) (2)

I have to (2) to express H_(n-1) and H_(n-2) in terms of derivatives of H_n, and substitute these into (1) to show that

H''_n - xH'_n + nH_n = 0 (3)

for n>= 0. Now let O_n(x) = exp(-(x^2)/4 )H_n(x). Use (3) to show that

O''_n +(n+1/2-(x^2)/3)O_n=0

Thank you.