Legendre polynomial : show by recurrence that $P_n(1) = 1$

Question

Starting from the recurrent relation of Legendre polynomial :

$(n+1)P_{n+1}(x) - (2n+1)xP_n(x) + nP_{n-1}(x) = 0 $ for $n \geq 1$ with $P_0(x) = 1$ and $P_1(x) = x$

How can I show by recurrence that $P_n(1) = 1$ ?

Answer 1

Simply: take any $n\in\Bbb N$, $n\ge 2$, and assume $P_{n}(1)=P_{n-1}(1)=1$. Then for $x=1$ we get $(n+1)P_{n+1}(1)-(2n+1)+n=0$, hence trivially $P_{n+1}(1)=1$ by the induction argument.