Sign Changes of Coefficients of Powers of the Infinite Borwein Product

Abstract

We denote by $c_t^{(m)}(n)$ the coefficient of $q^n$ in the series expansion of $(q;q{)}_{\infty }^m(q^t;q^t{)}_{\infty }^{-m}$, which is the $m$-th power of the infinite Borwein product. Let $t$ and $m$ be positive integers with $m(t-1)\le 24$. We provide asymptotic formula for $c_t^{(m)}(n)$, and give characterizations of $n$ for which $c_t^{(m)}(n)$ is positive, negative or zero. We show that $c_t^{(m)}(n)$ 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 $p$.

28 pages (including an Appendix with 5 pages). This paper has been rewritten after its first version (v1) with a different title. We found that the results in v1 can be generalized and thus rewrote it in v2. Minor changes will be made after v2. Please do not cite v1