Method of generating q -expansion coefficients for conformal block and N=2N=2 Nekrasov function by β-deformed matrix model

We observe that, at β  -deformed matrix models for the four-point conformal block, the point q=0q=0 is the point where the three-Penner type model becomes a pair of decoupled two-Penner type models and where, in the planar limit, (an array of), two-cut eigenvalue distribution(s) coalesce into (that of) one-cut one(s). We treat the Dotsenko–Fateev multiple integral, with their paths under the recent discussion, as perturbed double-Selberg matrix model (at q=0q=0, it becomes a pair of Selberg integrals) to construct two kinds of generating functions for the q  -expansion coefficients and compute some. A formula associated with the Jack polynomial is noted. The second Nekrasov coefficient for SU(2)SU(2) with Nf=4Nf=4 is derived. A pair of Young diagrams appears naturally. The finite N   loop equation at q=0q=0 as well as its planar limit is solved exactly, providing a useful tool to evaluate the coefficients as those of the resolvents. The planar free energy in the q-expansion is computed to the lowest non-trivial order. A free field representation of the Nekrasov function is given.