Sign changes of coefficients of powers of the infinite Borwein product

Abstract

We denote by ct(m)(n) the coefficient of qn in the series expansion of (q;q)∞m(qt;qt)∞−m, which is the m-th power of the infinite Borwein product. Let t and m be positive integers with m(t−1)≤24. We provide asymptotic formula for ct(m)(n), and give characterizations of n for which ct(m)(n) is positive, negative or zero. We show that ct(m)(n) is ultimately periodic in sign and conjecture that this is still true for other positive integer values of t and m. Furthermore, we confirm this conjecture in the cases (t,m)=(2,m),(p,1),(p,3) for arbitrary positive integer m and prime p.