  We consider expansions of functions in $L^{p}(\mathbb{R},|x|^{2k}dx)$, $1\leq
p<+\infty$ with respect to Dunkl-Hermite functions in the rank-one setting. We
actually define the heat-diffusion and Poisson integrals in the one-dimensional
Dunkl setting and study their properties. Next, we define and deal with Hilbert
transforms and conjugate Poisson integrals in the same setting. The formers
occur to be Calder\'on-Zygmund operators and hence their mapping properties
follow from general results.
