Applications of a new P-Q modular equation of degree two

• Published in: arXiv.org 2020-05
• Main Authors: Prabhakaran, D J, kumar, K Ranjith
• Format: Article
• Language: English
• Subjects: Formulas (mathematics); Invariants; Polynomials
• ISSN: 2331-8422

At scattered places in his first notebook, Ramanujan recorded the values for 107 class invariants or irreducible monic polynomials satisfied by them. On pages 294-299 in his second notebook, he gave a table of values for 77 class invariants \(G_n\) and \(g_n\) in his second notebook. Traditionally, \(G_n\) is determined for odd values of \(n\) and \(g_n\) for even values of \(n\). On pages 338 and 339 in his first notebook, Ramanujan defined the remarkable product of theta-functions \(a_{m, n}\). Also, he recorded eighteen explicit values depending on two parameters, namely, \(m\), and \(n\), where these are odd integers. In this paper, we initiate to study explicit evaluations of \(G_n\) for even values of \(n\). We establish a new general formula for the explicit evaluations of \(G_n\) involving class invariant \(g_n\). For this purpose, we derive a new P-Q modular equation of degree two. Further application of this modular equation, we establish a new formula to explicit evaluation of \(a_{m, 2}\). Also, we compute several explicit values of class invariant \(g_{n}\) and singular moduli \(\alpha_n\).