Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 17B / b / Solution

Suppose $m<n$. The Rodrigues formula and $n$ integrations by parts givebecause $p_m^{(n)}=0$. Every boundary term vanishes: the exponential controls infinity, while the remaining power of $x$ has positive exponent at zero. Symmetry of the inner product gives orthogonality whenever $m\ne n$.

For the norm, monicity gives $p_n^{(n)}=n!$, so the same calculation yields