We will show that
(
1
−
q
)
(
1
−
q
2
)
…
(
1
−
q
m
)
(1-q)(1-q^2)\dots (1-q^m)
is a polynomial in
q
q
with coefficients from
{
−
1
,
0
,
1
}
\{-1,0,1\}
iff
m
=
1
,
2
,
3
,
m=1,\ 2,\ 3,
or 5 and explore some interesting consequences of this result. We find explicit formulas for the
q
q
-series coefficients of
(
1
−
q
2
)
(
1
−
q
3
)
(
1
−
q
4
)
(
1
−
q
5
)
…
(1-q^2)(1-q^3)(1-q^4)(1-q^5)\dots
and
(
1
−
q
3
)
(
1
−
q
4
)
(
1
−
q
5
)
(
1
−
q
6
)
…
(1-q^3)(1-q^4)(1-q^5)(1-q^6)\dots
. In doing so, we extend certain observations made by Sudler in 1964. We also discuss the classification of the products
(
1
−
q
)
(
1
−
q
2
)
…
(
1
−
q
m
)
(1-q)(1-q^2)\dots (1-q^m)
and some related series with respect to their absolute largest coefficients.