Asymptotic expansions for partitions generated by infinite products

Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of partitions with parts in Λ ⊂ N Λ ⊂ N ( (Λ )=1 gcd ( Λ ) = 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 Λ Λ 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 so(5) so ( 5 ) . We also study the Witten zeta function ζ _so(5) ζ so ( 5 ) , which is of independent interest.