  Our main goal in this article is to study mixed norm estimates for the
Ces\`{a}ro means associated with Dunkl--Hermite expansions on $\mathbb{R}^d$.
These expansions arise when one consider the Dunkl--Hermite operator (or Dunkl
harmonic oscillator) $H_{\kappa}:=-\Delta_{\kappa}+|x|^2$, where
$\Delta_{\kappa}$ stands for the Dunkl--Laplacian. It is shown that the desired
mixed norm estimates are equivalent to vector-valued inequalities for a
sequence of Ces\`{a}ro means for Laguerre expansions with shifted parameter. In
order to obtain the latter, we develop an argument to extend these operators
for complex values of the parameters involved and apply a version of three
lines lemma.
