q-deformed Painlevé τ function and q-deformed conformal blocks

We propose the q-deformation of the Gamayun–Iorgov–Lisovyy formula for the Painlevé τ function. Namely, we propose the formula for the τ function for the q-difference Painlevé equation corresponding to the $A_7^{\left(1\right)\prime }$ surface (and the $A_1^{\left(1\right)}$ symmetry) in the Sakai classification. In this formula, the τ function equals the series of q-Virasoro Whittaker conformal blocks (equivalently, the Nekrasov partition functions for pure SU(2) 5d theory). (paper)