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We study five dimensional AGT correspondence by means of the q-deformedbeta-ensemble technique. We provide a special basis of states in the q-deformedCFT Hilbert space consisting of generalized Macdonald polynomials, derive theloop equations for the beta-ensemble and obtain the factorization formulas forthe corresponding matrix elements. We prove the spectral duality for Nekrasovfunctions and discuss its meaning for conformal blocks. We also clarify therelation between topological strings and q-Liouville vertex operators.

Generalized Macdonald polynomials, spectral duality for conformal blocks and AGT correspondence in five dimensions