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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2402.02546 |
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| _version_ | 1866910318597242880 |
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| author | Campbell, John M. |
| author_facet | Campbell, John M. |
| contents | Let $R(q)$ denote the Rogers-Ramanujan continued fraction for $|q| < 1$. By applying the RootApproximant command in the Wolfram language to expressions involving the theta function $f(-q) := (q;q)_{\infty}$ given in modular relations due to Yi, this provides a systematic way of obtaining experimentally discovered evaluations for $R\big(e^{-π\sqrt{r}}\big)$, for $r \in \mathbb{Q}_{> 0}$. We succeed in applying this approach to obtain explicit closed forms, in terms of radicals over $\mathbb{Q}$, for the Rogers-Ramanujan continued fraction that have not previously been discovered or proved. We prove our closed forms using the icosahedral equation for $R$ together with closed forms for and modular relations associated with Ramanujan's $G$- and $g$-functions. An especially remarkable closed form that we introduce and prove is for $R\big( e^{-π\sqrt{48/5} } \big)$, in view of the computational difficulties surrounding the application of an order-25 modular relation in the evaluation of $G_{48/5}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02546 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Applications of the icosahedral equation for the Rogers-Ramanujan continued fraction Campbell, John M. Number Theory 11F03 Let $R(q)$ denote the Rogers-Ramanujan continued fraction for $|q| < 1$. By applying the RootApproximant command in the Wolfram language to expressions involving the theta function $f(-q) := (q;q)_{\infty}$ given in modular relations due to Yi, this provides a systematic way of obtaining experimentally discovered evaluations for $R\big(e^{-π\sqrt{r}}\big)$, for $r \in \mathbb{Q}_{> 0}$. We succeed in applying this approach to obtain explicit closed forms, in terms of radicals over $\mathbb{Q}$, for the Rogers-Ramanujan continued fraction that have not previously been discovered or proved. We prove our closed forms using the icosahedral equation for $R$ together with closed forms for and modular relations associated with Ramanujan's $G$- and $g$-functions. An especially remarkable closed form that we introduce and prove is for $R\big( e^{-π\sqrt{48/5} } \big)$, in view of the computational difficulties surrounding the application of an order-25 modular relation in the evaluation of $G_{48/5}$. |
| title | Applications of the icosahedral equation for the Rogers-Ramanujan continued fraction |
| topic | Number Theory 11F03 |
| url | https://arxiv.org/abs/2402.02546 |