On Entry 8 of Chapter 19 of Ramanujan’s Second Notebook

K. R. Vasuki and A. Darshan
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 1, Pages 157–165
DOI: 10.7546/nntdm.2025.31.1.157-165
Full paper (PDF, 194 Kb)

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Authors and affiliations

K. R. Vasuki
Department of Studies in Mathematics, University of Mysore
Manasagangotri, Mysuru – 570 006, India

A. Darshan
Department of Mathematics, Government Science College – Autonomous
Hassan — 573 201, India

Abstract

The aim of this paper is to give an alternative proof of Entry 8 of Chapter 19 of Ramanujan’s Second Notebook. Further, we deduce certain modular equations of degree 5 as a consequence of Entry 8 of Chapter 19.

Keywords

  • Theta functions
  • Ramanujan’s _1\psi_1 summation formula
  • Modular equations

2020 Mathematics Subject Classification

  • 11F20
  • 33C05

References

  1. Bailey, W. N. (1952). A further note on two of Ramanujan’s formulae. The Quarterly Journal of Mathematics, 3(1), 158–160.
  2. Berndt, B. C. (1991). Ramanujan’s Notebooks: Part III. Springer Science & Business Media, New York.
  3. Cooper, S. (2017). Ramanujan’s Theta Functions. Springer, Cham.
  4. Kang, S.-Y. (1999). Some theorems on the Rogers–Ramanujan continued fraction and associated theta function identities in Ramanujan’s lost notebook. Ramanujan Journal, 3, 91–111.
  5. Ramanujan, S. (1957). Notebooks, Volume I & II. Tata Institute of Fundamental Research, Bombay.
  6. Ramanujan, S. (1988). The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi.

Manuscript history

  • Received: 2 August 2024
  • Revised: 26 April 2025
  • Accepted: 27 April 2025
  • Online First: 28 April 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Vasuki, K. R., & Darshan, A. (2025). On Entry 8 of Chapter 19 of Ramanujan’s Second Notebook. Notes on Number Theory and Discrete Mathematics, 31(1), 157-165, DOI: 10.7546/nntdm.2025.31.1.157-165.

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