= Normalization of the ordinary Laguerre polynomials
{title2=$\int_0^\infty e^{-x}L_n(x)^2\,dx=1$}
The <Laguerre polynomial> normalized by $L_n(0)=1$ has coefficients $L_n(x)=\sum_{j=0}^n(-1)^j\binom njx^j/j!$. By the <Leibniz rule>, this is equivalent to <Rodrigues' formula>, $e^{-x}L_n(x)=D^n(e^{-x}x^n)/n!$. For any <polynomial> $q$ of degree below $n$, $n$ applications of <integration by parts> give $\int_0^\infty e^{-x}L_nq\,dx=0$. All endpoint terms vanish: derivatives of $e^{-x}x^n$ of order below $n$ vanish at zero and decay at infinity. Since $L_n^{(n)}=(-1)^n$, the same calculation gives
$$
\int_0^\infty e^{-x}L_n^2\,dx
=\frac{(-1)^n}{n!}\int_0^\infty e^{-x}x^nL_n^{(n)}\,dx
=\frac{\Gamma(n+1)}{n!}=1.
$$
Thus these <orthogonal polynomials> are already an <orthonormal set> for the <weighted inner product> with weight $e^{-x}$.
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