I must show that the polynomial

satisfies a diferential equation.I used some notation(1,2..) so that I don't rewrite the same thing again. By diferentiating

and

using induction on n,I'm supposed to show that, for n >= 1,

(2)

I have to use (2) to express

and

in terms of derivatives of

, and substitute these into (1) to show that

(3)

for n>= 0. Now let

Using (3) I must show that

**Also H_n is a polynomial defined as follows:** and

; then, for n >=2, H_{n} is defined by the recurrence

(1)

**Attempt at a solution**
I need help with the entire problem mainly because for example

I can only prove it for the base case when but fot the general I get

=

......and I do not know what do do next...if it isn't obvious I just substituted in the relation that I have to prove since I'm assuming that everything from n-1 is true......

Also for the other parts I still need help in showing (3) and (4).Thanks in advance