  Andrews and Bressoud, Alladi and Gordon, and others, have proven, in a number
of papers, that the coefficients in various arithmetic progressions in the
series expansions of certain infinite $q$-products vanish. In the present paper
it is shown that these results follow automatically (simply by specializing
parameters) in an identity derived from a special case of Ramanujan's
$_1\psi_1$ identity.
  Likewise, a number of authors have proven results about the $m$-dissections
of certain infinite $q$-products using various methods. It is shown that many
of these $m$-dissections also follow automatically (again simply by
specializing parameters) from this same identity alluded to above.
  Two identities that mat be considered as extensions of two Identities of
Ramanujan are also derived.
  It is also shown how applying similar ideas to certain other Lambert series
gives rise to some rather curious $q$-series identities, such as, for any
positive integer $m$, \begin{multline*} {\displaystyle
\frac{\left(q,q,a,\frac{q}{a},\frac{b q}{d}, \frac{dq}{b}, \frac{aq}{b d},
\frac{b d q}{a};q\right)_{\infty }}
{\left(b,\frac{q}{b},d,\frac{q}{d},\frac{a}{b},\frac{bq}{a},\frac{a}{d},\frac{dq}{a};q\right)_{\infty
}}}
  = \sum _{r=0}^{m-1} q^r \frac{ \left(q^m,q^m,a q^{2 r},\frac{q^{m-2
r}}{a},\frac{b q^m}{d},\frac{d q^m}{b}, \frac{a q^m}{b d},\frac{b
dq^m}{a};q^m\right){}_{\infty }} {\left(b q^r,\frac{q^{m-r}}{b},d
q^r,\frac{q^{m-r}}{d},\frac{a q^r}{b},\frac{b q^{m-r}}{a},\frac{a q^r}{d},
\frac{dq^{m-r}}{a};q^m\right){}_{\infty }} \end{multline*}
  and
  \begin{equation*} (aq;q)_{\infty}\sum_{n=1}^{\infty} \frac{n a^n
q^{n}}{(q;q)_n} = \sum_{r=1}^{m}(aq^{r};q^m)_{\infty}\sum_{n=1}^{\infty}
\frac{na^n q^{n r}}{(q^m;q^m)_n}. \end{equation*} Applications to the Fine
function $F(a,b;t)$ are also considered.
