  Quite recently, the first author investigated vanishing coefficients of the
arithmetic progressions in several $q$-series expansions. In this paper, we
further study the signs of coefficients in two $q$-series expansions and
establish some arithmetic relations for several $q$-series expansions by means
of Ramanujan's theta functions. We obtain the 5-dissections of these two
$q$-series and give combinatorial interpretations for these dissections.
Moreover, we obtain four $q$-series identities involving the aforementioned
$q$-series, two of which were proved by Kim and Toh via modular forms.
