Note on two modular equations of Ramanujan

Dazhao Tang ()
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Dazhao Tang: Chongqing Normal University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 1, 47-53

Abstract: Abstract 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 $$f(-q)$$ f ( - q ) , its reciprocal, and Ramanujan’s theta function $$\psi (q)$$ ψ ( q ) . As by-products, we also establish four q-series identities concerning some specialized Jacobi theta series.

Keywords: Modular equations; Rogers–Ramanujan continued fraction; Dissections; Euler product; Theta series; 11B65; 11A55; 05A30; 14K25 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-022-00346-2

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