  Homogeneous Besov and Triebel-Lizorkin spaces associated with
multi-dimensional Laguerre function expansions of Hermite type with index
$\alpha \in [-1/2,\infty)^d\backslash (-1/2,1/2)^d$, $d\geq 1$, are defined and
investigated. To achieve expected goals Schwartz type spaces on $R^d_+$ are
introduced and then tempered type distributions are constructed. Also, ideas
from a recent paper of Bui and Duong on Besov and Triebel-Lizorkin spaces
associated with Hermite functions expansions are used. This means, in
particular, using molecular decomposition and an appropriate form of a
Calder\'on reproducing formula.
