  We show that the dual partition function of the pure $\mathcal N=2$ $SU(2)$
gauge theory in the self-dual $\Omega$-background (a) is given by Fredholm
determinant of a generalized Bessel kernel and (b) coincides with the tau
function associated to the general solution of the Painlev\'e III equation of
type $D_8$ (radial sine-Gordon equation). In particular, the principal minor
expansion of the Fredholm determinant yields Nekrasov combinatorial sums over
pairs of Young diagrams.
