Ratio asymptotics for orthogonal rational functions on an interval Original Research Article

Let {α1,α2,…} be a sequence of real numbers outside the interval [−1,1] and μ a positive bounded Borel measure on this interval satisfying the Erdős–Turán condition μ′>0 a.e., where μ′ is the Radon–Nikodym derivative of the measure μ with respect to the Lebesgue measure. 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 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.