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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 asymptoticbehavior of the sequences of monic polynomials that are orthogonal with respectto $w_{\lambda}(x;t)$, as well as the asymptotic behavior of the recurrencecoefficient, when the degree, or alternatively, the parameter $t$, tend toinfinity. We also investigate existence and uniqueness of positive solutions ofthe nonlinear difference equation satisfied by the recurrence coefficients andprove properties of the zeros of the generalized Freud polynomials.

Properties of Generalized Freud Polynomials