Recurrence and asymptotics for orthonormal rational functions on an interval

Let mu be a positive bounded Borel measure on a subset I of the real line and A = {alpha(1),...,alpha(n)} a sequence of arbitrary 'complex' poles outside I. Suppose {phi(1),...,phi(n)} is the sequence of rational functions with poles in A orthonormal on I with respect to mu. First, we are concerned with reducing the number of different coefficients in the three-term recurrence relation satisfied by these orthonormal rational functions. Next, we consider the case in which I = [-1, 1] and mu satisfies the Erdos-Turan condition mu' > 0 a.e. on I (where mu' is the Radon-Nikodym derivative of the measure mu with respect to the Lebesgue measure) to discuss the convergence of phi(n+1)(x)/phi(n)(x) as n tends to infinity and to derive asymptotic formulas for the recurrence coefficients in the three-term recurrence relation. Finally, we give a strong convergence result for phi(n)(x) under the more restrictive condition that mu satisfies the Szego condition (1 - x(2))(-1/2) log mu'(x) epsilon L(1)([-1, 1]).