  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.
