Generalized βand (q,t)-deformed partition functions with W-representations and Nekrasov partition functions

We construct the generalized beta and (q,t)-deformed partition functions through W representations,where the expansions are respectively with respect to the generalized Jack and Macdonald polynomials labeled by N-tuple of Young diagrams. We find that there are the profoundinterrelations between our deformed partition functions andthe 4dand 5dNekrasov partition functions. Since the corre-sponding Nekrasov partition functions can be given by vertex operators, the remarkable connection between our beta and(q,t)-deformed W-operators and vertex operators is revealed in this paper. In addition, we investigate the higher Hamilto-nians for the generalized Jack and Macdonald polynomials