On the Nonnegative Solution of a Freud Three-Term Recurrence

This article deals with the sequence ?={?n}n=0, 1, ? defined by the three-term recurrence n=4?n(?n?1+?n+?n+1), n=1, 2, ?, and by the initial conditions ?0=0, ?1=?(3/4)/?(1/4). Owing both to connections between the ?n's 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 ? has been studied by many authors. Properties of ? have been shown and computational algorithms provided. In this paper we show further properties of ?. First we establish bounds for the departure of ? from the sequence to which it asymptotically converges. Then we prove that ? is an increasing sequence.