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A remark on modular equations involving Rogers-Ramanujan continued fraction via $5$-dissections (2410.14149v1)
Published 18 Oct 2024 in math.NT and math.CO
Abstract: In this paper, we study the $5$-dissections of certain Ramanujan's theta functions, particularly $\psi(q)\psi(q2), \varphi(-q)$ and $\varphi(-q)\varphi(-q2)$, and derive an identity for $q(q;q){\infty}6/(q5;q5){\infty}6$ 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 $R(q), R(q2)$ and $R(q4)$, which was recorded by Ramanujan in his lost notebook, and establish modular equations involving $R(q), R(q2), R(q4), R(q8)$ and $R(q{16})$.