On certain new modular equations in the alternative bases analogous to those in the classical base

K R Vasuki; Shibu Andia

doi:10.1007/s44426-025-00010-2

Home
Search
On certain new modular equations in the alternative bases analogous to those in the classical base

https://doi.org/10.1007/s44426-025-00010-2

On certain new modular equations in the alternative bases analogous to those in the classical base

Jul 15, 2025
Acta Universitatis Sapientiae, Mathematica
K R Vasuki +1 more

Listen

Abstract
Literature Map
References
Similar Papers

Abstract

In the unorganized portion of his second notebook [6], Ramanujan has recorded modular equations corresponding to signatures 3 and 4. The proofs of the same can be found in Chapter 33 of [3]. In this article, we use the method of parametrization to establish a few new modular equations in signature 3 and 4 which are analogous to modular equations in signature 2 due to Ramanujan. Using the same method, we also prove a few modular equations in signatures 3 and 4, recorded by Ramanujan in his second notebook [6].

Similar Papers

Research Article
213
10.1090/s0002-9947-1995-1311903-0

Ramanujan’s theories of elliptic functions to alternative bases

Jan 01, 1995
Transactions of the American Mathematical Society
Bruce C Berndt + 2 more +2

In his famous paper on modular equations and approximations to π \pi , Ramanujan offers several series representations for 1 / π 1/\pi , which he claims are derived from "corresponding theories" in which the classical base q q is replaced by one of three other bases. The formulas for 1 / π 1/\pi were only recently proved by J. M. and P. B. Borwein in 1987, but these "corresponding theories" have never been heretofore developed. However, on six pages of his notebooks, Ramanujan gives approximately 50 results without proofs in these theories. The purpose of this paper is to prove all of these claims, and several further results are established as well.