Ratio Asymptotics for Orthogonal Rational Functions on the Interval [−1, 1]

Let {α1, α2, . . . } be a sequence of real numbers outside the interval [−1, 1] and μ a positive bounded Borel measure on this interval. We introduce rational functions φn(x) with poles {α1, . . . , αn} orthogonal on [−1, 1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of φn+1(x)/φn(x) as n tends to infinity under certain assumptions on the measure and the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three term recurrence relation satisfied by the orthonormal functions.