= Laguerre orthonormality
{c}
{title2=$\int_0^\infty L_n(x)L_m(x)e^{-x}\,dx=\delta_{nm}$}
The ordinary <Laguerre polynomials>, normalized by $L_n(0)=1$, are orthonormal for the weighted <inner product> displayed above. Their coefficient formula is $L_n(x)=\sum_{k=0}^n(-1)^k\binom nkx^k/k!$. To prove the relation, use <Rodrigues' formula> $e^{-x}L_n=(n!)^{-1}(x^ne^{-x})^{(n)}$ and integrate by parts $n$ times against $L_m$. For $n>m$ its $n$th derivative vanishes. At zero every derivative of $x^ne^{-x}$ of order less than $n$ vanishes, and at infinity the exponential kills every polynomial boundary term. For $n=m$, $L_n^{(n)}=(-1)^n$, so the integral is $(n!)^{-1}\int_0^\infty x^ne^{-x}\,dx=1$. Symmetry covers $m>n$. Thus these are orthonormal without an extra normalization factor.
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