Norms on polynomials

Note that $p_N$ has antiderivatives that vanish both at $-1$ and $1$. Assume (without loss of generality) that $N>M$; we can iterate integration by parts to get
$$ \langle p_N,p_M\rangle=(-1)^N\int_{-1}^1(x^2-1)^N\,\frac{d^{M+N}}{dx^{M+N}}(x^2-1)^M\,dx=0, $$ where the derivative is zero as the degree of $(x^2-1)^M$ is less than $N+M$.

When $N=M$, we need $$\frac{d^{2N}}{dx^{2N}}(x^2-1)^N.$$ The $2N$-derivative of a monic polynomial of degree $2N$ is the constant $(2N)!$. Thus (using Wolfram Alpha) $$ \langle p_N,p_N\rangle = (-1)^N\int_{-1}^1(x^2-1)^N\,(2N)!\,dx=(-1)^N\,(2N)! \frac{\sqrt\pi(-1)^N\Gamma(N+1)}{\Gamma(N+\tfrac32)}=\frac{\sqrt\pi\,(2N)!\,N!}{\Gamma(N+\tfrac32)} $$