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

Gu, Cheng-Yang ; Guo, Victor J. W.

Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 757-765.

Résumé

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.

Détail

DOI : 10.21136/CMJ.2020.0516-18

Classification : 05A10, 05A30, 11A07, 11B65
Keywords: congruences; $q$-binomial coefficient; cyclotomic polynomial; Franklin's involution