A note on the Borwein conjecture

Jiyou Li

Published 2015-12-03Version 1

The Borwein conjecture asserts that for any positive integer $n$ 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 note we prove that for any $k\leq n$, $$a_{3k}+a_{3(n+1)+3k}+\cdots+a_{3n(n+1)+3k}>0.$$