On Q-Real and Q-Complex Numbers

In arXiv:1812.00170, S. Morier-Genoud and V. Ovsienko introduced the notion of the q-rational number [x]_q, x∈ Q, a rational function specializing to x at q=1, obtained by q-deforming the continued fraction expansion of x. In arXiv:1908.04365 they introduced q-real numbers [x]_q, x∈ R - a Laurent series in q converging to the rational function [x]_q when x∈ Q. In arXiv:2102.00891 it is proved that if x∈ Q_>1 then the series [x]_q converges for |q|<3-2√(2)≈ 0.17 and conjectured that for all x∈ R_>1 this series converges in some disk centered in the origin, with the expected common radius of convergence R_*=3-√(5)/2≈ 0.38, achieved when x=1+√(5)/2 is the golden ratio. This was proved for rational x in arXiv:2405.15970 using the theory of Kleinian groups. In this paper we (partially) prove this conjecture by showing that for all x∈ R_>1, the series [x]_q converges in the disk |q|<3-2√(2) to a nonvanishing holomorphic function. This is achieved by giving an expansion of 1/[x]_q into a q-adically convergent series of rational functions converging absolutely and uniformly on compact sets in an explicit region D containing this disk. We also show that this expansion converges to a positive analytic function on the interval (-3-√(5)/2,1), giving a definition of [x]_q for q from this interval. Moreover, we show that the result of arXiv:2405.15970 implies convergence of [x]_q for |q|<2-√(3)≈ 0.27. We also give examples of explicit computation of [x]_q for transcendental numbers x, e.g. x= cotan(1). Finally, we propose a definition of the q-complex number [τ]_q, a meromorphic function of τ∈ C_+ which expresses via hypergeometric functions of modular functions of τ.