On the infinite Borwein product raised to a positive real power

TL;DR

This paper analyzes the coefficients of a generalized infinite Borwein product raised to a positive real power, providing asymptotic formulas, growth estimates, and partial confirmation of conjectured sign patterns, with implications for number theory.

Contribution

It introduces new asymptotic formulas and growth estimates for coefficients of the Borwein product, and confirms conjectured sign patterns for specific cases.

Findings

Asymptotic formula for coefficients using Hardy--Ramanujan--Rademacher method

Partial confirmation of sign pattern conjecture for p=3

Establishment of vanishing and divisibility properties

Abstract

In this paper, we study properties of the coefficients appearing in the $q$-series expansion of $\prod_{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $\delta$. We use the Hardy--Ramanujan--Rademacher circle method to give an asymptotic formula for the coefficients. For $p=3$ we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $\delta$ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $p=3$ case.