The space \((S^+)'\) of tempered distributions with positive support is the dual of the space \(S^+\) of restrictions to \([0,\infty)\) of rapidly decreasing functions. With \(L^{\alpha}_ n(t)=\sum^{n}_{k=0}\left( \begin{matrix} n+\alpha \\ n-k\end{matrix} \right)(-t)^ k/k!\) and \({\mathcal L}^{\alpha}_ n(t)=[n!/\Gamma (n+\alpha +1)]^{1/2}e^{- t/2}t^{\alpha /2}L^{\alpha}_ n(t)\quad (\alpha >-1),\) let \(\phi \in t^{\alpha /2}S^+\) and \(a_ n=\int^{\infty}_{0}\phi (t){\mathcal L}^{\alpha}_ n(t)dt,\) then \((a_ n)_ n\in s\), where s is the space of rapidly decreasing sequences, and \(\phi (t)=\sum^{\infty}_{n}a_ n{\mathcal L}^{\alpha}_ n(t)\). Conversely, given \((a_ n)_ n\in s\), there exists a \(\phi \in t^{\alpha /2}S^+\) such that \(a_ n=\int^{\infty}_{0}\phi (t){\mathcal L}^{\alpha}_ n(t)dt\) and \(t^{\alpha /2}S^+\) and s are isomorphic as topological vector spaces. [For \(\alpha =0\), these results were published in 1971, see M. Guillemot-Teissiers, Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 25, 519-573 (1971; Zbl 0225.46037).] It is also shown that the Hankel transform of degree \(\alpha\) is an isomorphism of \((t^{\alpha /2}S^+)'\) in \((t^{\alpha /2}S^+)'\), of \(t^{\alpha /2}S^+\) in \(t^{\alpha /2}S^+\), and that \({\mathcal H}^ 2_{\alpha}\) is the identity map. It also follows that \((S^+)'\) is a convolution algebra.