AI 中文总结
本文通过黎曼-希尔伯特问题构造了q-Painlevé VI方程的Tau函数为弗雷德霍姆行列式，揭示其消失条件，关联其与qPVI超越函数的关系，并推导了小时间下的渐近展开。
AI 中文摘要
我们通过与q-差分第六Painlevé方程（qPVI）相关的一般黎曼-希尔伯特问题，给出了其Tau函数的解析构造，将其表示为弗雷德霍姆行列式。我们证明该Tau函数在其定义域上是解析函数，且仅当对应的黎曼-希尔伯特问题在该点逐点不可解时，Tau函数才会在特定点消失。我们将对应的qPVI超越函数用该Tau函数及三个参数移位后的副本表示。这四个Tau函数中任意一个的消失，对应于qPVI初值空间上的超越函数取值于特定对应例外直线。最后，我们推导了小时间t下Tau函数的渐近展开。
英文摘要
We give an analytic construction of a tau function for the $q$-difference sixth Painlevé equation ($q$PVI) as a Fredholm determinant through the general Riemann-Hilbert problem associated with it. We show that the tau function is an analytic function on its domain of definition that vanishes at a particular point if and only if the corresponding Riemann-Hilbert problem is point-wise not solvable there. We express the corresponding $q$PVI transcendents in terms of the tau function as well as three copies of it with some of the parameters shifted. Then the vanishing of any of these four tau functions corresponds to the transcendents taking value in a specific corresponding exceptional line on the initial value space of $q$PVI. Finally, we derive an asymptotic expansion of the tau function for small times $t$.
Comments32 pages, 3 figures
