= Full-line Gaussian moment recurrence
{title2=$J_n=\int_{-\infty}^{\infty}x^ne^{-x^2}\,dx$}
For integer $n\geq2$, <integration by parts> gives $J_n=(n-1)J_{n-2}/2$. The boundary term $-x^{n-1}e^{-x^2}/2$ vanishes at both infinities because <Gaussian decay dominates every power>. The full-line initial values are $J_0=\sqrt\pi$ and $J_1=0$, the latter by odd symmetry. Hence
$$
J_{2m}=\frac{(2m-1)!!}{2^m}\sqrt\pi,\qquad J_{2m+1}=0.
$$
Unlike the half-line <Gaussian moment recurrence>, full-line odd moments vanish and even moments are twice their half-line values. The <domain of integration> must be retained when transferring a recurrence between the two settings.
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