5-Dissections and sign patterns of Ramanujan's parameter and its companion

In 1998, Michael Hirschhorn discovered 5-dissections of the Rogers–Ramanujan continued fraction $R(q)$ and its reciprocal. In this paper, we obtain the 5-dissections for functions $R(q)R(q^2{)}^2$ and $R(q{)}^2\mathrm{/}R(q^2)$, which are essentially Ramanujan’s parameter and its companion. 5-Dissections of the reciprocals of these two functions are derived as well. These 5-dissections imply that the coefficients in their series expansions have periodic sign patterns with few exceptions.