Let \(f(x)\) be a function such that \((1-x^2) f”(x) – 2x f'(x) + \alpha (\alpha+1) f(x) =0\)
for some \(\alpha \not\in \mathbb{N}\). Define \(P_n (x) = \frac{d^n}{dx^n} (x^2-1)^n\) for \(n =0,1,…\). Compute \(\int_{-1}^1 f(x) P_n(x) dx.\)