For $k\in \mathbb{N}$ let $$P_k(x)=\frac{1}{2^{k}k!}\frac{d^k}{dx^k}(x^2-1)^k$$ (1) Show that the $P_k$ are $L^2(-1,1)$ orthogonal.
I can't solve the integral with the definition given. I also tried to find the $k$-th polynomial using the binomial theorem, another dead end.
(2) Show that $P_k(1)=1$
(3) Show the three-term-recursion $$kP_k(x)=(2k-1)xP_{k-1}(x)-(k-1)P_{k-2}(x)$$
Again I don't know how this can be done without a more explicit representation of $P_k$. Is there even a more explicit representation of the polynomials? Or is some other trickery involved?
Edit: The final goal is to show these are the Legendre-polynomials.