标题： Riemann-Hilbert问题，Fredholm行列式，明确的组合展开式，以及一般$q$-Painlevé III$_3$tau函数的连接公式 显示英文标题

标题： Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general $q$-Painlevé III$_3$ tau functions

摘要： 我们将与类型$A_7^{(1)'}$的$q$-Painlevé 方程相对应的$q$-差分线性系统重新表述为圆上的 Riemann-Hilbert 问题。然后，我们考虑由这个 Riemann-Hilbert 问题的跳跃所构建的 Fredholm 表决式，并证明它满足等价于$P(A_7^{(1)'})$的双线性关系。 我们还以明确的因式分解形式找到了这个弗雷德霍姆行列式的极小子式展开，并证明它与$q$-变形共形块的傅里叶级数或者纯$5d$ $\mathcal{N}=1$ $SU(2)$ 规范理论的配分函数（包括带有Chern-Simons项的情形）相一致。最后，我们解决了这些等 mono-单值化tau函数的连接问题，从而找到了它们的全局行为。 显示英文摘要

摘要： We reformulate the $q$-difference linear system corresponding to the $q$-Painlev\'e equation of type $A_7^{(1)'}$ as a Riemann-Hilbert problem on a circle. Then, we consider the Fredholm determinant built from the jump of this Riemann-Hilbert problem and prove that it satisfies bilinear relations equivalent to $P(A_7^{(1)'})$. We also find the minor expansion of this Fredholm determinant in explicit factorized form and prove that it coincides with the Fourier series in $q$-deformed conformal blocks, or partition functions of the pure $5d$ $\mathcal{N}=1$ $SU(2)$ gauge theory, including the cases with the Chern-Simons term. Finally, we solve the connection problem for these isomonodromic tau functions, finding in this way their global behavior.