On radius of convergence of $q$-deformed real numbers
Abstract
We study analytic properties of ``-deformed real numbers'', a notion recently introduced by two of us. A -deformed positive real number is a power series with integer coefficients in one formal variable~. We study the radius of convergence of these power series assuming that is a complex variable. Our main conjecture, which can be viewed as a -analogue of Hurwitz's Irrational Number Theorem, claims that the -deformed golden ratio has the smallest radius of convergence among all real numbers. The conjecture is proved for certain class of rational numbers and confirmed by a number of computer experiments. We also prove the explicit lower bounds for the radius of convergence for the -deformed convergents of golden and silver ratios.
Keywords
Cite
@article{arxiv.2102.00891,
title = {On radius of convergence of $q$-deformed real numbers},
author = {Ludivine Leclere and Sophie Morier-Genoud and Valentin Ovsienko and Alexander Veselov},
journal= {arXiv preprint arXiv:2102.00891},
year = {2022}
}
Comments
16 pages, 1 figure