$q$-analogues of two supercongruences of Z.-W. Sun

@article{Gu2020,
abstract = {We give several different $q$-analogues of the following two congruences of Z.-W. Sun: \[ \sum \_\{k=0\}^\{(p^\{r\}-1)/2\}\frac\{1\}\{8^k\}\{2k\atopwithdelims ()k\} \equiv \Bigl (\frac\{2\}\{p^r\}\Bigr )\hspace\{10.0pt\}(\@mod \; p^2)\quad \text\{and\}\quad \sum \_\{k=0\}^\{(p^\{r\}-1)/2\}\frac\{1\}\{16^k\}\{2k\atopwithdelims ()k\}\equiv \Bigl (\frac\{3\}\{p^r\}\Bigr )\hspace\{10.0pt\}(\@mod \; p^2), \] where $p$ is an odd prime, $r$ is a positive integer, and $(\frac\{m\}\{n\})$ 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.},
author = {Gu, Cheng-Yang, Guo, Victor J. W.},
journal = {Czechoslovak Mathematical Journal},
keywords = {congruences; $q$-binomial coefficient; cyclotomic polynomial; Franklin’s involution},
language = {eng},
number = {3},
pages = {757-765},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$q$-analogues of two supercongruences of Z.-W. Sun},
url = {http://eudml.org/doc/297125},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Gu, Cheng-Yang
AU - Guo, Victor J. W.
TI - $q$-analogues of two supercongruences of Z.-W. Sun
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 3
SP - 757
EP - 765
AB - We give several different $q$-analogues of the following two congruences of Z.-W. Sun: \[ \sum _{k=0}^{(p^{r}-1)/2}\frac{1}{8^k}{2k\atopwithdelims ()k} \equiv \Bigl (\frac{2}{p^r}\Bigr )\hspace{10.0pt}(\@mod \; p^2)\quad \text{and}\quad \sum _{k=0}^{(p^{r}-1)/2}\frac{1}{16^k}{2k\atopwithdelims ()k}\equiv \Bigl (\frac{3}{p^r}\Bigr )\hspace{10.0pt}(\@mod \; p^2), \] where $p$ is an odd prime, $r$ is a positive integer, and $(\frac{m}{n})$ 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.
LA - eng
KW - congruences; $q$-binomial coefficient; cyclotomic polynomial; Franklin’s involution
UR - http://eudml.org/doc/297125
ER -