We apply results of Garsia-Haiman-Tesler on Macdonald polynomials to the problem of computation of integrals of tautological classes over the Hilbert schemes of surfaces, studied by Marian-Oprea-Pandharipande. Using localization, these results allow us to find new functional equations for the generating series of integrals. MOP paper considers two kind of integrals: the so-called Chern integrals resp. Verlinde integrals. The answer to the problem is encoded in series A1, A2, A3, A4, A5 resp. B1, B2, B3, B4. All the series except A4, A5, B3, B4 were computed in MOP and a conjecture motivated by mathematical physics was formulated relating A4 to B3 and A5 to B4. It was also conjectured that A4, A5, B3, B4 are algebraic functions. Solving our functional equations we prove the former conjecture and obtain explicit formulas for A4 and B3, thus proving a part of the latter conjecture. We also give a conjectural formula for A5 and B4. This is a joint work with Lothar Göttsche.