  We consider the semi-classical generalized Freud weight function
\[w_{\lambda}(x;t) = |x|^{2\lambda+1}\exp(-x^4 +tx^2),\qquad x\in\mathbb{R},\]
with $ \lambda>-1$ and $t\in\mathbb{R}$ parameters. We analyze the asymptotic
behavior of the sequences of monic polynomials that are orthogonal with respect
to $w_{\lambda}(x;t)$, as well as the asymptotic behavior of the recurrence
coefficient, when the degree, or alternatively, the parameter $t$, tend to
infinity. We also investigate existence and uniqueness of positive solutions of
the nonlinear difference equation satisfied by the recurrence coefficients and
prove properties of the zeros of the generalized Freud polynomials.
