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arXiv · 2004.05854

The Explicit Evaluations Formula for Ramanujan's Singular Moduli and Ramanujan- Selberg Continued Fraction

D. J. Prabhakaran·K. Ranjith kumar
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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 employing 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 class invariant. For this purpose, we establish several new $P-Q$ mixed modular equations involving theta-functions. Further application of these modular equations, we derive a new formula to explicit evaluation of Ramanujan- Selberg continued fraction.

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D. J. Prabhakaran, K. Ranjith kumar. 2020-04-13. The Explicit Evaluations Formula for Ramanujan's Singular Moduli and Ramanujan- Selberg Continued Fraction. https://arxiv.org/abs/2004.05854

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