Mixed type multiple orthogonal polynomials for two Nikishin systems

TL;DR

This paper investigates the asymptotic behavior of mixed type multiple orthogonal polynomials derived from Nikishin systems, using potential theory and complex analysis to describe their logarithmic and ratio asymptotics.

Contribution

It introduces a new framework combining type I and type II multiple orthogonal polynomials within Nikishin systems and characterizes their asymptotics via equilibrium problems and Riemann surface conformal mappings.

Findings

Logarithmic asymptotics expressed through vector equilibrium solutions

Ratio asymptotics described by conformal maps of Riemann surfaces

Unified treatment of mixed type multiple orthogonal polynomials

Abstract

We study the logarithmic and ratio asymptotic of linear forms constructed from a Nikishin system which satisfy orthogonality conditions with respect to a system of measures generated from another Nikishin system. This construction combines type I and type II multiple orthogonal polynomials. The logarithmic asymptotic of the linear forms is expressed in terms of the extremal solution of an associated vector valued equilibrium problem for the logarithmic potential. The ratio asymptotic is described by means of a conformal representation of an appropriate Riemann surface of genus zero onto the extended complex plane.