Theta function identities involving fourth power

Praveenkumar, Siddaraju, R. Rangarajan
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 683–688
DOI: 10.7546/nntdm.2025.31.4.683-688
Full paper (PDF, 184 Kb)

Details

Authors and affiliations

Praveenkumar
Department of Mathematics, Government College for Women (Autonomous)
Mandya – 571401, India

Siddaraju
Department of Mathematics, Government First Grade College for Women
Chamarajanagara – 571313, India

R. Rangarajan
Department of studies in Mathematics, University of Mysore, Manasagangotri
Mysuru – 570006, India

Abstract

On page 241 of his Second Notebook, Ramanujan recorded one of his theta function identity, which involves the ratio of the fourth power of theta functions with respect to \psi(q). In this article, we give a new proof for this theta function identity. Also, we give a new proof of another identity with respect to \varphi(q) established by B. C. Berndt and we establish two new theta function identities analogous to Ramanujan’s theta function identities.

Keywords

  • Theta functions
  • Bailey’s summation formula

2020 Mathematics Subject Classification

  • 11F20
  • 33C20

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-Verlag, New York.
  3. Ramanujan, S. (1957). Notebooks (2 Volumes). Tata Institute of Fundamental Research, Bombay.
  4. Vasuki, K. R., & Veeresha, R. G. (2016). Ramanujan’s Eisenstein series of level 7 and 14. Journal of Number Theory, 159, 59–75.

Manuscript history

  • Received: 2 March 2025
  • Revised: 6 October 2025
  • Accepted: 8 October 2025
  • Online First: 10 October 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

Praveenkumar, Siddaraju, & Rangarajan, R. (2025). Theta function identities involving fourth power. Notes on Number Theory and Discrete Mathematics, 31(4), 683-688, DOI: 10.7546/nntdm.2025.31.4.683-688.

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