Asymptotic expansions for partitions generated by infinite products

Abstract

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