Method of Generating q-Expansion Coefficients for Conformal Block and N=2 Nekrasov Function by beta-Deformed Matrix Model

We observe that, at beta-deformed matrix models for the four-point conformal block, the point q=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=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) with N_f =4 is derived. A pair of Young diagrams appears naturally. The finite N loop equation at q=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.