Rodrigues' formula and Legendre’s polynomials Let $$P_n(z)=\frac{1}{n!\space2^n}\frac{d^n}{dz^n}(z^2-1)^n$$(i) Show that this is a polynomial of order n.
I can see that this is Rodrigues' formula, which means that it is a Legendre polynomial.  I do not see however, a rigorous way to prove that is a polynomial besides that statement.
(ii) Show that $$P_n(z) = \frac{1}{2^{n+1}\pi i}\int_C\frac{(s^2-1)^n}{(s-z)^{n+1}}ds,  n=0,1,2,...$$ where C is any positively-oriented simple closed contour surrounding the point z.
This part I am struggling to even begin.  I thought it might have something to do with the binomial formula, but that has led me nowhere.

Since $(s^2 - 1)^n$ is analytic inside and on $C$, and $z$ lies inside $C$, Cauchy's differentiation formula gives 
$$\frac{1}{2^{n+1}\pi i} \int_C \frac{(s^2 - 1)^n}{(s - z)^{n+1}}\, ds = \frac{1}{2^{n+1}\pi i} \frac{2\pi i}{n!} \frac{d^n}{ds^n}\bigg|_{s = z} (s^2  - 1)^n = P_n(z)$$
