Orthogonal polynomials and Padé approximants associated with a system of arcs

AbstractWe study the convergence behavior of the diagonal sequence of the Padé table associated with a function with branch points. Given a set of even number, 2l, of the branch points, a unique set S is constructed which consists of a number of analytic Jordan arcs ending at the branch points. We assume that these arcs are nonintersecting. Let σ(t) be a complex, never vanishing function defined on S, satisfying a Lipschitz type smoothness condition there, and let X(t) be the monic polynomial of degree 2l with zeros at the branch points. We construct orthogonal polynomials with respect to the weight function X+−(12)(t)σ(t) and study their asymptotic behavior. The orthogonal polynomials are defined without complex conjugation and the domain of integration is S. Some properties of these polynomials yield convergence in capacity of the diagonal sequence of Padé approximants to f(t) = ∫sdt′X+−(12)(t′)σ(t′ − t)−1 in any closed, bounded domain of the complex plain cut along S