Past exam of the mathematics course of the University of Cambridge / 2017 / ib / Paper 4 / 5A / iii

Evaluate $R_n$ at $x=1$ using $(x^2-1{)}^n=(x-1{)}^n(x+1{)}^n$. In the Leibniz rule for the $n$th derivative, every term vanishes at $1$ except the term putting all $n$ derivatives on $(x-1{)}^n$. Therefore $R_n(1)=n!2^n$. Combining this with $P_n(1)=1$ gives the normalisation in the Rodrigues formula:For $n=0$, the formula gives ${\alpha }_0=1$, as required. As an independent check, the leading term of $R_n$ is $(2n)!x^n/n!$, so the resulting leading coefficient of $P_n$ is $(2n)!/(2^n(n!{)}^2)$, consistent with the Legendre polynomial recurrence relation.