i have this problem that goes: show that

$\displaystyle (n+1)p_{n+1} (x)-xp_{n+1}^{'} = x(np_{n} (x)-xp_{n}^{'}(x))-p_{n-1}$

where

$\displaystyle (n+1)p_{n+1}(x)=(2n+1)xp_{n}(x)-np_{n-1}(x).................(a)$

i recalled a relation that says $\displaystyle p_{n+1}^{'}-p_{n-1}^{'}=(2n+1)p_{n}$

and i substituted it in equation (a) above,which gives

$\displaystyle (n+1)p_{n+1}(x)=x(p_{n+1}^{'}-p_{n-1}^{'})-np_{n-1}(x)$

rearranging, i have

$\displaystyle (n+1)p_{n+1}(x)-xp_{n+1}^{'}=-xp_{n-1}^{'}-np_{n-1}(x)$

there exit a relation also of this kind: $\displaystyle p_{n}^{'}=xp_{n-1}^{'}+np_{n-1}$

substituting, the equation now becomes

$\displaystyle (n+1)p_{n+1}(x)-xp_{n+1}^{'}=-p_{n}$

pls i need help at this point to enable me know where i got it wrong. thanks