Normalization of generalized Laguerre polynomials ($L_n^{(m)}(0)=\left(\genfrac{}{}{0ex}{}{n+m}{n}\right),\Vert L_n^{(m)}{\Vert }_{x^m{e}^{-x}}^2=\frac{(n+m)!}{n!}$)

For integers $m,n\ge 0$, the standard Rodrigues' formula $L_n^{(m)}=e^x{x}^{-m}{D}^n(x^{n+m}{e}^{-x})/n!$ 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 $\left(\genfrac{}{}{0ex}{}{n+m}{n}\right)$ and its squared norm by the square of that factor. Only when $m=0$ or $n=0$ do these normalizations already coincide.