Proof of a supercongruence conjecture of (F.3) of Swisher using the WZ-method
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2020
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913423508373504 |
|---|---|
| author | Jana, Arijit Kalita, Gautam |
| author_facet | Jana, Arijit Kalita, Gautam |
| contents | For a non-negative integer $m$, let $S(m)$ denote the sum given by $$S(m):=\sum_{n=0}^{m}\frac{(-1)^n(8n+1)}{n!^3}\left(\frac{1}{4}\right)_n^3.$$ Using the powerful WZ-method, for a prime $p\equiv 3$ $($mod $4)$ and an odd integer $r>1$, we here deduce a supercongruence relation for $S\left(\frac{p^r-3}{4}\right)$ in terms of values of $p$-adic gamma function. As a consequence, we prove one of the supercongruence conjectures of (F.3) posed by Swisher. This is the first attempt to prove supercongruences for a sum truncated at $\frac{p^r-(d-1)}{d}$ when $p^r\equiv -1$ $($mod $d)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2011_02762 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Proof of a supercongruence conjecture of (F.3) of Swisher using the WZ-method Jana, Arijit Kalita, Gautam Number Theory 11Y55, 11A07, 11B65, 40G99 For a non-negative integer $m$, let $S(m)$ denote the sum given by $$S(m):=\sum_{n=0}^{m}\frac{(-1)^n(8n+1)}{n!^3}\left(\frac{1}{4}\right)_n^3.$$ Using the powerful WZ-method, for a prime $p\equiv 3$ $($mod $4)$ and an odd integer $r>1$, we here deduce a supercongruence relation for $S\left(\frac{p^r-3}{4}\right)$ in terms of values of $p$-adic gamma function. As a consequence, we prove one of the supercongruence conjectures of (F.3) posed by Swisher. This is the first attempt to prove supercongruences for a sum truncated at $\frac{p^r-(d-1)}{d}$ when $p^r\equiv -1$ $($mod $d)$. |
| title | Proof of a supercongruence conjecture of (F.3) of Swisher using the WZ-method |
| topic | Number Theory 11Y55, 11A07, 11B65, 40G99 |
| url | https://arxiv.org/abs/2011.02762 |