1. ## Proof using deravatives.

Hi,
Wasn't too sure where to post this so feel free to move.
If
$\displaystyle \int x^MK(x)dx\not=0$
and
$\displaystyle \int x^mK(x)dx=0$
where m = 1,2,3......M-1
Show that the formula
$\displaystyle K_{[M+2]}(x)=\frac{3}{2}K_{[M]}(x)+\frac{1}{2}xK'_{[M]}(x)$.
Produces K of order M+2
and now
$\displaystyle \int x^{M+2}K(x)dx\not=0$

Any help appreciated cos I don't have a clue where to begin.

2. Originally Posted by markrvr
Hi,
Wasn't too sure where to post this so feel free to move.
If
$\displaystyle k_M=\int t^MKtdt\not=0$
and
$\displaystyle \int t^mK(t)dt=0$
where m = 1,2,3......M-1
Show that the formula
$\displaystyle K_{[M+2]}(x)=\frac{3}{2}K_{[M]}(x)+\frac{1}{2}xK'_{[M]}(x)$.
Produces K of order M+2
Any help appreciated cos I don't have a clue where to begin.
Your notation is a bit confusing. Is the "$\displaystyle K_{[M]}$" the same as the "$\displaystyle k_M$" in your first equation?

3. Sorry, no they are different. That notation was from a different part I forgot to delete it.