Properties of Generalized Freud Polynomials

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.