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