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In 1998, Michael Hirschhorn discovered 5-dissections of the Rogers--Ramanujancontinued fraction $R(q)$ and its reciprocal. In this paper, we obtain the5-dissections for functions $R(q)R(q^2)^2$ and $R(q)^2/R(q^2)$, which areessentially Ramanujan's parameter and its companion. 5-Dissections of thereciprocals of these two functions are derived as well. These 5-dissectionsimply that the coefficients in their series expansions have periodic signpatterns with few exceptions.

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