Generalized $$\beta $$ β and (q, t)-deformed partition functions with W-representations and Nekrasov partition functions

Abstract

Abstract We construct the generalized $$\beta $$ β 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 profound interrelations between our deformed partition functions and the 4d and 5d Nekrasov partition functions. Since the corresponding Nekrasov partition functions can be given by vertex operators, the remarkable connection between our $$\beta $$ β and (q, t)-deformed W-operators and vertex operators is revealed in this paper. In addition, we investigate the higher Hamiltonians for the generalized Jack and Macdonald polynomials.