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 \(f(-q)\), its reciprocal, and Ramanujan’s theta function \(\psi (q)\). As by-products, we also establish four q-series identities concerning some specialized Jacobi theta series.
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Acknowledgements
The author would like to acknowledge the anonymous referee for his/her careful reading and helpful suggestions that have improved the quality of this paper. This work was partially supported by the National Natural Science Foundation of China (No. 12201093), the Natural Science Foundation Project of Chongqing CSTB (No. CSTB2022NSCQ–MSX0387), the Science and Technology Research Program of Chongqing Municipal Education Commission (No. KJQN202200509) and the Doctoral start-up research grant (No. 21XLB038) of Chongqing Normal University.
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Communicated by B. Sury.
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Tang, D. Note on two modular equations of Ramanujan. Indian J Pure Appl Math 55, 47–53 (2024). https://doi.org/10.1007/s13226-022-00346-2
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DOI: https://doi.org/10.1007/s13226-022-00346-2