JOURNAL ARTICLE

Analogue of Ramanujan's function k(τ) for the cubic continued fraction.

  • Published In: International Journal of Number Theory, 2023, v. 19, n. 9. P. 2101 1 of 3

  • Database: Academic Search Ultimate 2 of 3

  • Authored By: Park, Yoon Kyung 3 of 3

Abstract

We study the modularity of the function u (τ) = C (τ) C (2 τ) , where C (τ) is Ramanujan's cubic continued fraction. It is an analogue of Ramanujan's function k (τ) = r (τ) r (2 τ) 2 , where r (τ) is the Rogers–Ramanujan continued fraction. We first prove the modularity of u (τ) and express C (τ) and C (2 τ) in terms of u (τ). Subsequently, we find modular equations of u (τ) of level n for every positive integer n by using affine models of modular curves. Finally, we demonstrate that the value of u (τ) generates the ray class field over an imaginary quadratic field modulo 2 for some τ in an imaginary quadratic field. [ABSTRACT FROM AUTHOR]

Additional Information

  • Source:International Journal of Number Theory. 2023/10, Vol. 19, Issue 9, p2101
  • Document Type:Article
  • Subject Area:History
  • Publication Date:2023
  • ISSN:1793-0421
  • DOI:10.1142/S1793042123501026
  • Accession Number:171874640
  • Copyright Statement:Copyright of International Journal of Number Theory is the property of World Scientific Publishing Company and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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