Scalar approaches to the limit distribution of the zeros of Hermite-Pade polynomials for a Nikishin system

The problem of the existence of a limit distribution of the zeros of Hermite-Padé polynomials for a pair of functions forming a Nikishin system is discussed. Two new scalar methods are proposed for the investigation of this problem. The first is based on a potential-theoretic equilibrium problem stated on a two-sheeted Riemann surface and on the use of the Gonchar-Rakhmanov-Stahl (GRS-)method in treating this problem. The second method is based on the existence of a three-sheeted Riemann surface with Nuttall partition into sheets which is associated with a given pair of functions f, f², and it uses only the maximum principle for subharmonic functions. The connection of these methods and the results obtained with Stahl's methods and results of 1987-88 is discussed. Results of numerical experiments are presented. Bibliography: 109 titles.