On $q$-real and $q$-complex numbers

Abstract

In arXiv:1812.00170, S. Morier-Genoud and V. Ovsienko introduced the notion of the $q$-rational number $[x]_q$, $x\in \Bbb Q$, 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 $[x]_q$, $x\in \Bbb R$ - a Laurent series in $q$ converging to the rational function $[x]_q$ when $x\in \Bbb Q$. In arXiv:2102.00891 it is proved that if $x\in \Bbb Q_{>1}$ then the series $[x]_q$ converges for $|q|<3-2\sqrt{2}\approx 0.17$ and conjectured that for all $x\in \Bbb R_{>1}$ this series converges in some disk centered in the origin, with the expected common radius of convergence $R_*=\frac{3-\sqrt{5}}{2}\approx 0.38$, achieved when $x=\frac{1+\sqrt{5}}{2}$ 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 $x\in \Bbb R_{>1}$, the series $[x]_q$ converges in the disk $|q|<3-2\sqrt{2}$ 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 $(-\frac{3-\sqrt{5}}{2},1)$, 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 $|q|<2-\sqrt{3}\approx 0.27$. We also give examples of explicit computation of $[x]_q$ for transcendental numbers $x$, e.g. $x={\rm cotan}(1)$. Finally, we propose a definition of the $q$-complex number $[\tau]_q$, a meromorphic function of $\tau\in \Bbb C_+$ which expresses via hypergeometric functions of modular functions of $\tau$.