= Normalization of generalized Laguerre polynomials
{title2=$L_n^{(m)}(0)=\binom{n+m}{n},\quad\|L_n^{(m)}\|_{x^me^{-x}}^2=\frac{(n+m)!}{n!}$}
For integers $m,n\geq0$, the standard <Rodrigues' formula> $L_n^{(m)}=e^xx^{-m}D^n(x^{n+m}e^{-x})/n!$ has the endpoint value and squared norm displayed above. The lowest-power term after $n$ differentiations is $(n+m)!x^m/m!$, giving the endpoint value after division by $n!x^m$. For the norm, integrate by parts $n$ times against the same <polynomial>; its leading coefficient is $(-1)^n/n!$, so the final <derivative> is $(-1)^n$. All boundary terms vanish and the remaining <Gamma function> integral is $(n+m)!/n!$. Normalizing the value at zero to one divides the <polynomial> by $\binom{n+m}{n}$ and its squared norm by the square of that factor. Only when $m=0$ or $n=0$ do these normalizations already coincide.
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