A note on the Borwein conjecture

Abstract

A conjecture of Borwein asserts that for any positive integers $n$ and $k$, the coefficient $a_{3k}$ of $q^{3k}$ in the expansion of ${\prod }_{j=0}^n(1-q^{3j+1})(1-q^{3j+2})$ is nonnegative. In this paper we prove that for any $0\le k\le n$, there is a constant $0<c<1$ such that $$a_{3k}+a_{3(n+1)+3k}+\cdots +a_{3n(n+1)+3k}=\frac{2\cdot 3^n}{n+1}(1+O(c^n)).$$ In particular, $$a_{3k}+a_{3(n+1)+3k}+\cdots +a_{3n(n+1)+3k}>0.$$

13 pages; proofs simplified and details added

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