Ramanujan's continued fractions of order $10$ as modular functions
| Autor: | Aricheta, Victor Manuel, Guadalupe, Russelle |
|---|---|
| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We explore the modularity of the continued fractions $I(\tau), J(\tau), T_1(\tau), T_2(\tau)$ and $U(\tau)=I(\tau)/J(\tau)$ of order $10$, where $I(\tau)$ and $J(\tau)$ 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 $\eta$-quotient $g(\tau)$ that generates the field of all modular functions on the congruence subgroup $\Gamma_0(10)$. Consequently, we prove that modular equations for $g(\tau)$ and $U(\tau)$ exist at any level, and derive these equations of prime levels $p\leq 11$. We also show that the above continued fractions of order $10$ can be explicitly evaluated using a singular value of $g(\tau)$, which, under certain conditions, generates the Hilbert class field of an imaginary class field. We employ the methods of Lee and Park to establish our results. Comment: 23 pages. arXiv admin note: text overlap with arXiv:2306.08782 |
| Databáze: | arXiv |
| Externí odkaz: |
|