Asymptotic expansions for partitions generated by infinite products

Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in $$ ( $$ ) and good analytic properties of the corresponding zeta function, generalizing work of Meinardus. In this paper, we extend their work to prove asymptotic formulas if $$ is a multiset of integers and the zeta function has multiple poles. In particular, our results imply an asymptotic formula for the number of irreducible representations of degree n of $$ . We also study the Witten zeta function $$ , which is of independent interest.