On the Nonnegative Solution of a Freud Three-Term Recurrence Original Research Article

This article deals with the sequence ξ={ξn}n=0, 1, … defined by the three-term recurrence n=4ξn(ξn−1+ξn+ξn+1), n=1, 2, …, and by the initial conditions ξ0=0, ξ1=Γ(3/4)/Γ(1/4). Owing both to connections between the ξnʹs and orthonormal polynomials with respect to the weight function w: w(x)=exp(−x4) and to difficulties that arise when one attempts to compute its elements, the sequence ξ has been studied by many authors. Properties of ξ have been shown and computational algorithms provided. In this paper we show further properties of ξ. First we establish bounds for the departure of ξ from the sequence to which it asymptotically converges. Then we prove that ξ is an increasing sequence.