For \alpha>0 we consider the system \ell_k^{(\alpha-1)/2}(x) of the Laguerre
functions which are eigenfunctions of the differential operator Lf
=-\frac{d^2}{dx^2}f - \frac{\alpha}{x}\frac{d}{dx}f + x^2 f. We define an
atomic Hardy space H^1_{at}(X), which is a subspace of L^1((0,\infty), x^\alpha
dx). Then we prove that the space H^1_{at}(X) is also characterized by the
Riesz transform Rf =\sqrt{\pi} \dx L^{-1/2} f in the sense that f\in
H^1_{at}(X) if and only if f, Rf \in L^1((0,\infty),x^\alpha dx).