Properties of generalized Freud polynomials

We consider the semiclassical generalized Freud weight function wλ(x;t)=|x|2λ+1exp(−x4+tx2),λ>−1,x∈R.We analyse the asymptotic behaviour of the sequences of monic polynomials that are orthogonal with respect to wλ(x;t), as well as the asymptotic behaviour 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 discrete equation satisfied by the recurrence coefficients and prove properties of the zeros of the generalized Freud polynomials.