We define the inner product on the space $\Bbb R[x]$ by $$\langle P,Q\rangle=\int_{-1}^1P(x)Q(x)(1-x^2)^\alpha dx$$ where $\alpha>-1$. I need to prove that for all $n\in\mathbb N$ $$\frac{d^n}{dx^n}((1-x^2)^{\alpha+n})=(1-x^2)^\alpha J_n^\alpha(x)$$ where $J_n^\alpha$ is a polynomial with degree $n$ and we have $$\langle J_n^\alpha,J_m^\alpha\rangle=0\qquad n\ne m$$ and determine $J_n^\alpha(1)$ and $J_n^\alpha(-1)$.
What I have tried so far: for the first question I used the Leibniz theorem for the derivative $$\frac{d^n}{dx^n}((1-x^2)^{\alpha+n})=\frac{d^n}{dx^n}((1-x)^{\alpha+n}(1+x)^{\alpha+n})$$ but I can't prove that the degree of $J_n^\alpha$ is $n$ and I'm entirely stuck on the other questions. Any suggestions would be appreciated.
Everything is there(p23):
http://www-user.tu-chemnitz.de/~peju/skripte/orthopol/OrthPoly_Engl.pdf
You just have to particularize for $\alpha=\beta$