标题： 关于$q$-实数和$q$-复数 显示英文标题

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

摘要： 在arXiv:1812.00170中，S. Morier-Genoud和V. Ovsienko引入了$q$有理数$[x]_q$，$x\in \Bbb Q$，一种有理函数，在$q=1$处退化为$x$，通过$q$变形$x$的连分数展开得到。 在arXiv:1908.04365中他们引入了$q$-实数$[x]_q$，$x\in \Bbb R$-在$q$中的一个洛朗级数当$x\in \Bbb Q$时收敛到有理函数$[x]_q$。 在 arXiv:2102.00891 中证明，如果$x\in \Bbb Q_{>1}$，则级数$[x]_q$对$|q|<3-2\sqrt{2}\approx 0.17$收敛，并猜想对于所有$x\in \Bbb R_{>1}$，该级数在原点为中心的某个圆盘内收敛，预期的共同收敛半径为$R_*=\frac{3-\sqrt{5}}{2}\approx 0.38$，在$x=\frac{1+\sqrt{5}}{2}$为黄金分割比时达到。这在 arXiv:2405.15970 中通过克莱因群理论对有理数$x$予以证明。 在本文中，我们通过证明对于所有$x\in \Bbb R_{>1}$，级数$[x]_q$在圆盘$|q|<3-2\sqrt{2}$内收敛到一个不恒为零的全纯函数，从而（部分）证明了这个猜想。 这是通过将$1/[x]_q$展开为一个$q$有理函数的级数，该级数在包含该圆盘的显式区域$D$内绝对且一致收敛来实现的。 我们还证明了此展开在区间$(-\frac{3-\sqrt{5}}{2},1)$上收敛于一个正的解析函数，由此给出了$[x]_q$对于$q$的定义。 此外，我们证明了 arXiv:2405.15970 的结果意味着$[x]_q$对于$|q|<2-\sqrt{3}\approx 0.27$的收敛性。 我们还给出了对超越数$x$的$[x]_q$的显式计算示例，例如 $x={\rm cotan}(1)$. 最后，我们提出一个$q$-复数$[\tau]_q$的定义，这是一个$\tau\in \Bbb C_+$的亚纯函数，它通过模函数的超几何函数来表示$\tau$。 显示英文摘要

摘要： 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 $\tau$.