  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 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 $\beta$ 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.
