On the Nonnegative Solution of a Freud Three-Term Recurrence

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