Asymptotics of Hermite-Padé polynomials for a set of functions with different branch points

Abstract

For an example where the functions have different branch points we derive the asymptoties of diagonal Hermite-Pad~ polynomials of type I. The method uses an integral equation obtained by approximating a reproducing kernel. The results are consistent with a new conjecture on the asymptotics of the polynomials associated with more general functions with different branch points.

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

“…Next, the function g n is meromorphic on R 2 and the genus of the Riemann surface R 2 is zero, and hence the function g n is completely defined by its divisor (38) (of the zeros and poles). As a result, from (38) we have the following explicit representation for the function g n :…”

“…At present, the answer to the problem of the limit distribution of the zeros of Hermite-Padé polynomials is available only for some particular classes of analytic functions (see [17], [34], [38], [19], [2], [4], [40], [32]). As a rule, the limit distribution of the zeros of Hermite-Padé polynomials for a pair of functions f 1 , f 2 can be described following the approach first proposed by Nuttall (see [36], [38]) in terms related to some three-sheeted Riemann surface which in a certain sense 1 is "associated" with the pair of functions f 1 , f 2 (for the relation between the three-sheeted Riemann surface with the asymptotics of Hermite-Padé polynomials, see also [25], [6], [26].) For a pair of functions f 1 , f 2 of form (1) the above problem was solved by Nikishin [34] in 1986 (see also [33], [35], [7]).…”

Nuttall's integral equation and Bernshtein's asymptotic formula for a complex weight

“…[2]- [4]), но и на случай полиномов Эрмита-Паде (см. [5], ср. [6]), а также на случай переменных ("variable", т. е. зависящих от номера многочлена) весов [7], [8].…”

Распределение Нулей Полиномов Эрмита - Паде Для Пары Функций, Образующей Систему Никишина

“…также [2; § 3, п. 3.1] и [6]) в связи с изучением сильной асимптотики полиномов Эрмита-Паде первого рода. А именно, для набора из m функций, независимых и мероморфных на m-листной римановой поверхности (в обозначениях Наттолла здесь мы рассматриваем три функции: f 0 (z) ≡ 1, f 1 и f 2 , и, тем самым, m = 3) φ(z (3)…”

“…В определенном смысле настоящая работа продолжает исследование таких полиномов, начатое в работах Е. М. Никишина [16] и Дж. Наттолла [5], [2], [6], но для пары функций. Отметим, что для пары функций, образующей систему Анжелеско (т.е.…”