Lacunarity of Han–Nekrasov–Okounkov q-Series

Abstract

AbstractA power series is called lacunary if “almost all” of its coefficients are zero. Integer partitions have motivated the classification of lacunary specializations of Han’s extension of the Nekrasov–Okounkov formula. More precisely, we consider the modular forms $$\begin{aligned}F_{a,b,c}(z) :=\frac{\eta (24az)^a \eta (24acz)^{b-a}}{\eta (24z)},\end{aligned}$$ F a , b , c ( z ) : = η ( 24 a z ) a η ( 24 a c z ) b - a η ( 24 z ) , defined in terms of the Dedekind $$\eta $$ η -function, for integers $$a,c \ge 1$$ a , c ≥ 1 , where $$b \ge 1$$ b ≥ 1 is odd throughout. Serre (Publications Mathématiques de l’IHÉS 123–201:2959–2968, 1981) determined the lacunarity of the series when $$a = c = 1$$ a = c = 1 . Later, Clader et al. (Am Math Soc 137(9):2959–2968, 2009) extended this result by allowing a to be general and completely classified the $$F_{a,b,1}(z)$$ F a , b , 1 ( z ) which are lacunary. Here, we consider all c and show that for $${a \in \{1,2,3\}}$$ a ∈ { 1 , 2 , 3 } , there are infinite families of lacunary series. However, for $$a \ge 4$$ a ≥ 4 , we show that there are finitely many triples (a, b, c) such that $$F_{a,b,c}(z)$$ F a , b , c ( z ) is lacunary. In particular, if $$a \ge 4$$ a ≥ 4 , $$b \ge 7$$ b ≥ 7 , and $$c \ge 2$$ c ≥ 2 , then $$F_{a,b,c}(z)$$ F a , b , c ( z ) is not lacunary. Underlying this result is the proof the t-core partition conjecture proved by Granville and Ono (Trans Am Math Soc 348(1):331–347, 1996).