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

2020/04/13 by D. J. Prabhakaran, Prabhakaran, D. J., K. Ranjith kumar +1
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2004.05854

openalex publication_date 2020/04/13 · openalex created_date 2020/04/17 · openalex updated_date 2026/07/28

Abstract

At scattered places of his notebooks, Ramanujan recorded over 30 values of singular moduli α_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 α_9n and α_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.

Related

• P--Q "mixed" modular equations of degree 15
• The explicit evaluations of Ramanujan's remarkable product of theta-functions a_{m, n}
• Applications of a new P-Q modular equation of degree two
• Analytical Evaluation of Ramanujan Series for 1/π via Degree-2, Degree-3, Degree-7, and Degree-19 Modular Transformations
• Generalizations of Ramanujans Continued fractions