Extension of a conjectural supercongruence of (G.3) of Swisher using Zeilberger's algorithm
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2020
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917716729790464 |
|---|---|
| author | Jana, Arijit Kalita, Gautam |
| author_facet | Jana, Arijit Kalita, Gautam |
| contents | Using Zeilberger's algorithm, we here give a proof of the supercongruence $$ \sum_{n=0}^{\frac{p^r-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^4}{(1)_n^4}\equiv -p^3 \sum_{n=0}^{\frac{p^{r-2}-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^4}{(1)_n^4} ~~(\text{mod }p^{\frac{3r-1}{2}}),$$ for any odd integer $r>3$. This extends the third conjectural supercongruence of (G.3) of Swisher to modulo higher prime powers than that expected by Swisher. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_02757 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Extension of a conjectural supercongruence of (G.3) of Swisher using Zeilberger's algorithm Jana, Arijit Kalita, Gautam Number Theory 11Y55, 11A07, 11B65, 40G99 Using Zeilberger's algorithm, we here give a proof of the supercongruence $$ \sum_{n=0}^{\frac{p^r-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^4}{(1)_n^4}\equiv -p^3 \sum_{n=0}^{\frac{p^{r-2}-3}{4}}(8n+1)\frac{\left(\frac{1}{4}\right)_n^4}{(1)_n^4} ~~(\text{mod }p^{\frac{3r-1}{2}}),$$ for any odd integer $r>3$. This extends the third conjectural supercongruence of (G.3) of Swisher to modulo higher prime powers than that expected by Swisher. |
| title | Extension of a conjectural supercongruence of (G.3) of Swisher using Zeilberger's algorithm |
| topic | Number Theory 11Y55, 11A07, 11B65, 40G99 |
| url | https://arxiv.org/abs/2011.02757 |