Czechoslovak Mathematical Journal, Vol. 70, No. 3, pp. 757-765, 2020
$q$-analogues of two supercongruences of Z.-W. Sun
Cheng-Yang Gu, Victor J. W. Guo
Received November 26, 2018. Published online January 22, 2020.
Abstract: We give several different $q$-analogues of the following two congruences of Z.-W. Sun:
$\sum_{k=0}^{(p^r-1)/2}\frac1{8^k}{2k\choose k} \equiv\Bigl(\frac2{p^r}\Bigr)\pmod{p^2}$ and $\sum_{k=0}^{(p^r-1)/2}\frac1{16^k}{2k\choose k}\equiv\Bigl(\frac3{p^r}\Bigr)\pmod{p^2}$,
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.
Affiliations: Cheng-Yang Gu, Victor J. W. Guo (corresponding author), School of Mathematics and Statistics, Huaiyin Normal University, Huai'an 223300, Jiangsu, P. R. China, e-mail: 525290408@qq.com, jwguo@hytc.edu.cn