Note on two modular equations of Ramanujan

In his notebooks and lost notebook, Ramanujan recorded two modular equations involving the Rogers–Ramanujan continued fraction. These two modular equations were subsequently proved by several scholars. In this paper, we provide another proof for these two modular equations in terms of the 5-dissections of the Euler product f(-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(-q)$$\end{document}, its reciprocal, and Ramanujan’s theta function ψ(q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (q)$$\end{document}. As by-products, we also establish four q-series identities concerning some specialized Jacobi theta series.