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

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