Asymptotic expansions for partitions generated by infinite products

Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in $\Lambda \subset N$ ( $gcd\left(\Lambda \right)=1$ ) 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 $\Lambda$ 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 $\mathrm{so}\left(5\right)$ . We also study the Witten zeta function ${\zeta }_{\mathrm{so}\left(5\right)}$ , which is of independent interest.