A Unified Transformation Formula for Ramanujan's Theta Function
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917076451459072 |
|---|---|
| author | Gurram, Mahipal |
| author_facet | Gurram, Mahipal |
| contents | In this paper, we derive a unified generalization of Ramanujan's transformation identities for the theta function $f(a,b)$, originally appearing in Ramanujan's Notebooks, Parts~III and IV. Using an approach based on residue-class dissections and modular substitutions, we obtain a closed transformation formula for $f(ζa, ζb)$, where $ζ$ is a primitive root of unity $m$. As special cases, we recover and systematically prove Ramanujan's classical results for $m=2,3,$ and $4$, including even odd dissections, cubic transformation and the compact quartic form involving complex coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09565 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Unified Transformation Formula for Ramanujan's Theta Function Gurram, Mahipal General Mathematics 11F27(Primary) In this paper, we derive a unified generalization of Ramanujan's transformation identities for the theta function $f(a,b)$, originally appearing in Ramanujan's Notebooks, Parts~III and IV. Using an approach based on residue-class dissections and modular substitutions, we obtain a closed transformation formula for $f(ζa, ζb)$, where $ζ$ is a primitive root of unity $m$. As special cases, we recover and systematically prove Ramanujan's classical results for $m=2,3,$ and $4$, including even odd dissections, cubic transformation and the compact quartic form involving complex coefficients. |
| title | A Unified Transformation Formula for Ramanujan's Theta Function |
| topic | General Mathematics 11F27(Primary) |
| url | https://arxiv.org/abs/2511.09565 |