Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general q-Painlevé III₃ tau functions

We reformulate the q-difference linear system corresponding to the q-Painlevé equation of type A^(1)′₇ 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^(1)′₇). 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 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.