| Title:
|
$q$-analogues of two supercongruences of Z.-W. Sun (English) |
| Author:
|
Gu, Cheng-Yang |
| Author:
|
Guo, Victor J. W. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
757-765 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We give several different $q$-analogues of the following two congruences of \hbox {Z.-W. Sun}: $$ \sum _{k=0}^{(p^{r}-1)/2}\frac {1}{8^k}{2k\choose k} \equiv \Bigl (\frac {2}{p^r}\Bigr )\pmod {p^2}\quad \text {and}\quad \sum _{k=0}^{(p^{r}-1)/2}\frac {1}{16^k}{2k\choose k}\equiv \Bigl (\frac {3}{p^r}\Bigr )\pmod {p^2}, $$ where $p$ is an odd prime, $r$ is a positive integer, and $(\frac mn)$ is the Jacobi symbol. The proofs of them require the use of some curious $q$-series identities, two of which are related to Franklin's involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012. (English) |
| Keyword:
|
congruences |
| Keyword:
|
$q$-binomial coefficient |
| Keyword:
|
cyclotomic polynomial |
| Keyword:
|
Franklin's involution |
| MSC:
|
05A10 |
| MSC:
|
05A30 |
| MSC:
|
11A07 |
| MSC:
|
11B65 |
| idZBL:
|
07250687 |
| idMR:
|
MR4151703 |
| DOI:
|
10.21136/CMJ.2020.0516-18 |
| . |
| Date available:
|
2020-09-07T09:37:57Z |
| Last updated:
|
2022-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148326 |
| . |
| Reference:
|
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