Rodrigues' formula proof

Question

Consider the polynomials $P_s (x) = \frac{1}{2^ss!}\frac{d^s}{dx^s}[(x^2-1)^s]$. Prove $\int_{-1}^{1}x^kP_s(x)dx = 0$ for all $k \in \mathbb{N}, k<s$.

Answer 1

The Rodrigues formula isn't the most helpful starting-point; we should instead use another definition of the $P_n$ viz. $P_n(1)=1,\,m\ne n\implies\int_{-1}^1P_mP_ndx=0$. We can prove by induction that $P_n$ is a unique degree-$n$ polynomial orthogonal to the polynomials $P_0,\,\cdots,\,P_{n-1}$. Since these $n$ polynomials are orthogonal, they comprise a basis of the $n$-dimensional space of polynomials of degree less than $n$. Therefore, if $k<n$ some linear combination of them is $x^k$, giving $\int_1^1 x^kP_n dx=0$. So the real question is why the family of polynomials thus defined satisfies the Rodrigues formula. You can find the proof in many places, including this video.