Monic Legendre polynomials (which are orthogonal polynomials) on $[-1,1]$ are defined as follows:
$$p_{0}(x)=1, \ p_{1}(x)=x$$
and $p_{n}(x)$ is a monic polynomial of degree $n$ such that $$\int_{-1}^{1} p_{i}(x) p_{j}(x) d x=0$$ for all $i \neq j$. We have to show that these polynomials satisfy $$p_{n+1}(x)=x p_{n}(x)-\left(\frac{n^{2}}{4 n^{2}-1}\right) p_{n-1}(x)$$
I tried to prove it by induction. The base case ($n=2$) is easy. I assumed that it is true for $n=k$. For $n=k+1$, as they are monic Legendre polynomial, so, $p_{k+1} = x^{k+2}+a_{k+1}x^{k+1}+a_kx^k+\dots +a_0.$ Then I tried to use the inner product condition i.e., $$\int_{-1}^{1} p_{k+2}(x) p_{i}(x) d x=0 \ \text{for}\ i= 0,1,2,\dots, k+1.$$ But it is getting very tedious because the integration depends on whether $k$ is even or odd. So, can anyone please help me with this in the right direction.
Thanks!