The following was a problem in a recent numerical analysis exam:
Let $k \in \mathbb{N}\setminus\{0\}$. Prove or disprove: $$ \int_{-1}^{1} cos\left(k \operatorname{arccos}(x)\right) \cdot \left(\frac{\partial^{k+1}}{\partial x^{k+1}}(x^2 - 1)^{k+1} \right) dx = 0 $$
Our professor said this was obviously true, since the term on the left is a Chebyshev polynomial and the term on the right is a Legendre polynomial. I know that Legendre polynomials are orthogonal with respect to the $L^2$ inner product (when $-1 \leq x \leq 1$), but what is the relation here? Is there any reason for using partials in the Legendre polynomials here (besides trying to confuse your students)?
The Legendre polynomial $P_{k}=C_{k}\frac{d^{k}}{dx^{k}}(x^{2}-1)^{k}$ ($C_k$ is a normalization constant) is orthogonal to all polynomials of lower order with respect to the inner product $(f,g)=\int_{-1}^{1}f(x)g(x)dx$. In fact, the Legendre polynomials can be derived by applying the Graham-Schmidt procedure to $\{ 1,x,x^{2},x^{3},\cdots \}$. So your Professor is correct.