A note on the Borwein conjecture

TL;DR

This paper investigates Borwein's conjecture, proving a sum of certain coefficients in a product expansion is positive and approaches a specific value exponentially fast as n increases.

Contribution

The paper provides a proof that sums of specific coefficients are positive and asymptotically approximate a scaled exponential function, advancing understanding of Borwein's conjecture.

Findings

Sum of coefficients approaches a scaled exponential as n increases

Sum of coefficients is strictly positive for all relevant k

Provides asymptotic formula with exponential convergence

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 $0 \leq k\leq n$, 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, $$a_{3k}+a_{3(n+1)+3k}+\cdots+a_{3n(n+1)+3k}>0.$$