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Ramanujan's continued fractions of order 10 as modular functions.

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Title: Ramanujan's continued fractions of order 10 as modular functions.
Authors: Aricheta, Victor Manuel1 (AUTHOR) vmaricheta@math.upd.edu.ph, Guadalupe, Russelle1 (AUTHOR) rguadalupe@math.upd.edu.ph
Source: Journal of Number Theory. Jan2026, Vol. 278, p214-244. 31p.
Subjects: Continued fractions, Modular functions, Equations
Abstract: We explore the modularity of the continued fractions I (τ) , J (τ) , T 1 (τ) , T 2 (τ) , and U (τ) = I (τ) / J (τ) of order 10, which are special cases of certain identities of Ramanujan. The continued fractions I (τ) and J (τ) were recently introduced by Rajkhowa and Saikia. We show that these continued fractions can be expressed in terms of an η -quotient g (τ) that generates the field of all modular functions on the congruence subgroup Γ 0 (10). Consequently, we deduce that the modular equations for g (τ) and U (τ) exist at any level and derive these equations of prime levels p ≤ 11. We also show that the continued fractions of order 10 can be explicitly evaluated using a singular value of g (τ) , which under certain conditions generates the Hilbert class field of an imaginary quadratic field. We employ the methods of Lee and Park to establish our results. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Number Theory is the property of Academic Press Inc. 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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  Data: Ramanujan's continued fractions of order 10 as modular functions.
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  Data: <searchLink fieldCode="AR" term="%22Aricheta%2C+Victor+Manuel%22">Aricheta, Victor Manuel</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> vmaricheta@math.upd.edu.ph</i><br /><searchLink fieldCode="AR" term="%22Guadalupe%2C+Russelle%22">Guadalupe, Russelle</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> rguadalupe@math.upd.edu.ph</i>
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  Data: <searchLink fieldCode="JN" term="%22Journal+of+Number+Theory%22">Journal of Number Theory</searchLink>. Jan2026, Vol. 278, p214-244. 31p.
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  Data: <searchLink fieldCode="DE" term="%22Continued+fractions%22">Continued fractions</searchLink><br /><searchLink fieldCode="DE" term="%22Modular+functions%22">Modular functions</searchLink><br /><searchLink fieldCode="DE" term="%22Equations%22">Equations</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: We explore the modularity of the continued fractions I (τ) , J (τ) , T 1 (τ) , T 2 (τ) , and U (τ) = I (τ) / J (τ) of order 10, which are special cases of certain identities of Ramanujan. The continued fractions I (τ) and J (τ) were recently introduced by Rajkhowa and Saikia. We show that these continued fractions can be expressed in terms of an η -quotient g (τ) that generates the field of all modular functions on the congruence subgroup Γ 0 (10). Consequently, we deduce that the modular equations for g (τ) and U (τ) exist at any level and derive these equations of prime levels p ≤ 11. We also show that the continued fractions of order 10 can be explicitly evaluated using a singular value of g (τ) , which under certain conditions generates the Hilbert class field of an imaginary quadratic field. We employ the methods of Lee and Park to establish our results. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Journal of Number Theory is the property of Academic Press Inc. 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.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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      – Type: doi
        Value: 10.1016/j.jnt.2025.04.001
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      – Code: eng
        Text: English
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        PageCount: 31
        StartPage: 214
    Subjects:
      – SubjectFull: Continued fractions
        Type: general
      – SubjectFull: Modular functions
        Type: general
      – SubjectFull: Equations
        Type: general
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      – TitleFull: Ramanujan's continued fractions of order 10 as modular functions.
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            NameFull: Aricheta, Victor Manuel
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            NameFull: Guadalupe, Russelle
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            – D: 01
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              Text: Jan2026
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              Y: 2026
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              Value: 278
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            – TitleFull: Journal of Number Theory
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