Asymptotic expansions for partitions generated by infinite products

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