  An attempt is made to formulate Gaiotto's S-duality relations in an explicit
quantitative form. Formally the problem is that of evaluation of the Racah
coefficients for the Virasoro algebra, and we approach it with the help of the
matrix model representation of the AGT-related conformal blocks and Nekrasov
functions. In the Seiberg-Witten limit, this S-duality reduces to the Legendre
transformation. In the simplest case, its lifting to the level of Nekrasov
functions is just the Fourier transform, while corrections are related to the
beta-deformation. We calculate them with the help of the matrix model approach
and observe that they vanish for beta=1. Explicit evaluation of the same
corrections from the U_q(sl(2)) infinite-dimensional representation formulas
due to B.Ponsot and J.Teshner remains an open problem.
