Modularity of certain products of the Rogers-Ramanujan continued fraction
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866908541458055168 |
|---|---|
| author | Guadalupe, Russelle |
| author_facet | Guadalupe, Russelle |
| contents | We study the modularity of the functions of the form $r(τ)^ar(2τ)^b$, where $a$ and $b$ are integers with $(a,b)\neq (0,0)$ and $r(τ)$ is the Rogers-Ramanujan continued fraction, which may be considered as companions to the Ramanujan's function $k(τ)=r(τ)r(2τ)^2$. In particular, we show that under some condition on $a$ and $b$, there are finitely many such functions generating the field of all modular functions on the congruence subgroup $Γ_1(10)$. Furthermore, we establish certain arithmetic properties of the function $l(τ)=r(2τ)/r(τ)^2$, which can be used to evaluate these products. We employ the methods of Lee and Park, and some properties of $η$-quotients and generalized $η$-quotients to prove our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06678 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modularity of certain products of the Rogers-Ramanujan continued fraction Guadalupe, Russelle Number Theory 11F03, 11F11, 11R37 We study the modularity of the functions of the form $r(τ)^ar(2τ)^b$, where $a$ and $b$ are integers with $(a,b)\neq (0,0)$ and $r(τ)$ is the Rogers-Ramanujan continued fraction, which may be considered as companions to the Ramanujan's function $k(τ)=r(τ)r(2τ)^2$. In particular, we show that under some condition on $a$ and $b$, there are finitely many such functions generating the field of all modular functions on the congruence subgroup $Γ_1(10)$. Furthermore, we establish certain arithmetic properties of the function $l(τ)=r(2τ)/r(τ)^2$, which can be used to evaluate these products. We employ the methods of Lee and Park, and some properties of $η$-quotients and generalized $η$-quotients to prove our results. |
| title | Modularity of certain products of the Rogers-Ramanujan continued fraction |
| topic | Number Theory 11F03, 11F11, 11R37 |
| url | https://arxiv.org/abs/2405.06678 |