Ramanujan's continued fractions of order $10$ as modular functions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Aricheta, Victor Manuel Guadalupe, Russelle |
| author_facet | Aricheta, Victor Manuel Guadalupe, Russelle |
| contents | 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 $10$, where $I(τ)$ and $J(τ)$ are introduced by Rajkhowa and Saikia, which are special cases of certain identities of Ramanujan. In particular, we show that these 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 prove that modular equations for $g(τ)$ and $U(τ)$ exist at any level and derive these equations of prime levels $p\leq 11$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05756 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Ramanujan's continued fractions of order $10$ as modular functions Aricheta, Victor Manuel Guadalupe, Russelle Number Theory Primary 11F03, 11F11, 11R37 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 $10$, where $I(τ)$ and $J(τ)$ are introduced by Rajkhowa and Saikia, which are special cases of certain identities of Ramanujan. In particular, we show that these 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 prove that modular equations for $g(τ)$ and $U(τ)$ exist at any level and derive these equations of prime levels $p\leq 11$. 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. |
| title | Ramanujan's continued fractions of order $10$ as modular functions |
| topic | Number Theory Primary 11F03, 11F11, 11R37 |
| url | https://arxiv.org/abs/2404.05756 |