Asymptotics of Hermite–Padé Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0)

We investigate the asymptotic behavior for type II Hermite–Padé approximation to two functions, where each function has two branch points and the pairs of branch points are separated. We give a classification of the cases such that the limiting counting measures for the poles of the Hermite–Padé approximants are described by an algebraic function h of order and genus 0. This situation gives rise to a vector-potential equilibrium problem for measures λ, μ1, and μ2, and the poles of the common denominator are asymptotically distributed like λ/2. We also work out the strong asymptotics for the corresponding Hermite–Padé approximants by using a 3 × 3 Riemann–Hilbert problem that characterizes this Hermite–Padé approximation problem.