  In this note we consider the maximal function for the generalized
Ornstein-Uhlenbeck semigroup in $\RR$ associated with the generalized Hermite
polynomials $\{H_n^{\mu}\}$ and prove that it is weak type (1,1) with respect
to $d\lambda_{\mu}(x) = |x|^{2\mu}e^{-|x|^2} dx,$ for $\mu >-1/2$ as well as
bounded on $L^p(d\lambda_\mu) $ for $p>1$
