Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 17B (Numerical Analysis)

By the Leibniz rule,After multiplication by $(-1{)}^n{x}^{-1/2}{e}^x$, the term $j=0$ is $x^n$, and every other term has lower degree. Thus $p_n$ is a monic polynomial of degree $n$. Direct calculation givesThese are the monic normalization of the generalized Laguerre polynomials with parameter $1/2$.

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

For monic orthogonal polynomials, taking the inner product of the recurrence with $p_{n-1}$ givesUsing part (b) and the Gamma function recurrence,

The Rodrigues expansion also shows that the coefficient of $x^{n-1}$ in $p_n$ isComparing the coefficients of $x^n$ intherefore givesand hence