Abstract
At scattered places of his notebooks, Ramanujan recorded over 30 values of singular moduli \(\alpha_n\). All those results were proved by Berndt et. al by using Weber–Ramanujan’s class invariants. In this paper, we initiate to derive the explicit evaluations formula for \(\alpha_{9n}\) and \(\alpha_{n/9}\) by involving the class invariant. For this purpose, we establish several new \(P-Q\) mixed modular equations involving theta-functions. We apply these modular equations further, deriving a new formula for the explicit evaluation of the Ramanujan–Selberg continued fraction.
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Prabhakaran, D.J., Ranjithkumar, K. Explicit Evaluation Formula for Ramanujan’s Singular Moduli and Ramanujan–Selberg Continued Fractions. Math Notes 110, 363–374 (2021). https://doi.org/10.1134/S0001434621090066
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DOI: https://doi.org/10.1134/S0001434621090066