| Title:
|
A $q$-congruence for a truncated $_{4}\varphi _{3}$ series (English) |
| Author:
|
Guo, Victor J. W. |
| Author:
|
Wei, Chuanan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
1157-1165 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\Phi _n(q)$ denote the $n$th cyclotomic polynomial in $q$. Recently, Guo, Schlosser and Zudilin proved that for any integer $n>1$ with $n\equiv 1\pmod {4}$, $$ \sum _{k=0}^{n-1}\frac {(q^{-1};q^2)_k^2(q^{-2};q^4)_k}{(q^2;q^2)_k^2 (q^4;q^4)_k}q^{6k} \equiv 0\pmod {\Phi _n(q)^2}, $$ where $(a;q)_m=(1-a)(1-aq)\cdots (1-aq^{m-1})$. In this note, we give a generalization of the above $q$-congruence to the modulus $\Phi _n(q)^3$ case. Meanwhile, we give a corresponding $q$-congruence modulo $\Phi _n(q)^2$ for $n\equiv 3\pmod {4}$. Our proof is based on the `creative microscoping' method, recently developed by Guo and Zudilin, and a $_4\varphi _3$ summation formula. (English) |
| Keyword:
|
basic hypergeometric series |
| Keyword:
|
Watson's transformation |
| Keyword:
|
$q$-congruence |
| Keyword:
|
supercongruence |
| Keyword:
|
creative microscoping |
| MSC:
|
11A07 |
| MSC:
|
11B65 |
| MSC:
|
33D15 |
| idZBL:
|
Zbl 07442481 |
| idMR:
|
MR4339118 |
| DOI:
|
10.21136/CMJ.2021.0317-20 |
| . |
| Date available:
|
2021-11-08T16:04:43Z |
| Last updated:
|
2024-01-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149245 |
| . |
| Reference:
|
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| Reference:
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| Reference:
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