On one example of a Nikishin system

TL;DR

This paper presents an example of a Markov function forming a Nikishin system with its powers and conjectures a broader generalization for all natural numbers.

Contribution

It provides a specific example of a Markov function creating a Nikishin system with its powers and proposes a conjecture for the existence of such functions for all natural numbers.

Findings

An example of a Markov function forming a Nikishin system with its powers.

A conjecture that such functions exist for all natural numbers.

Discussion of the properties of the constructed example.

Abstract

The paper puts forward an example of a~Markov function $f=\operatorname{const}+\widehat{\sigma}$ such that the three functions $f,f^2$ and $f^3$ form a Nikishin system. A conjecture is proposed that there exists a~Markov function $f$ such that, for each $n\in\mathbb N$, the system $f,f^2,\dots,f^n$ constitutes a~Nikishin system. Bibliography:~20~titles.