A remark on modular equations involving Rogers-Ramanujan continued fraction via $5$-dissections

In this paper, we study the $5$-dissections of certain Ramanujan’s theta functions, particularly $\psi (q)\psi (q^2),\phi (-q)$ and $\phi (-q)\phi (-q^2)$, and derive an identity for $q(q;q{)}_{\mathrm{\infty }}^6\mathrm{/}(q^5;q^5{)}_{\mathrm{\infty }}^6$ in terms of certain products of the Rogers-Ramanujan continued fraction $R(q)$. Using this identity, we give another proof of the modular equation involving $R(q),R(q^2)$ and $R(q^4)$, which was recorded by Ramanujan in his lost notebook, and establish modular equations involving $R(q),R(q^2),R(q^4),R(q^8)$ and $R(q^{16})$.