Geodesic
$q$-analogues of two supercongruences of Z.-W. Sun
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 757-765

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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.
DOI : 10.21136/CMJ.2020.0516-18
Classification : 05A10, 05A30, 11A07, 11B65
Keywords: congruences; $q$-binomial coefficient; cyclotomic polynomial; Franklin's involution 
@article{10_21136_CMJ_2020_0516_18,
     author = {Gu, Cheng-Yang and Guo, Victor J. W.},
     title = {$q$-analogues of two supercongruences of {Z.-W.} {Sun}},
     journal = {Czechoslovak Mathematical Journal},
     pages = {757--765},
     publisher = {mathdoc},
     volume = {70},
     number = {3},
     year = {2020},
     doi = {10.21136/CMJ.2020.0516-18},
     mrnumber = {4151703},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0516-18/}
}
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
SP  - 757
EP  - 765
VL  - 70
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0516-18/
DO  - 10.21136/CMJ.2020.0516-18
LA  - en
ID  - 10_21136_CMJ_2020_0516_18
ER  - 
%0 Journal Article
%A Gu, Cheng-Yang
%A Guo, Victor J. W.
%T $q$-analogues of two supercongruences of Z.-W. Sun
%J Czechoslovak Mathematical Journal
%D 2020
%P 757-765
%V 70
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0516-18/
%R 10.21136/CMJ.2020.0516-18
%G en
%F 10_21136_CMJ_2020_0516_18
Gu, Cheng-Yang; Guo, Victor J. W. $q$-analogues of two supercongruences of Z.-W. Sun. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 757-765. doi: 10.21136/CMJ.2020.0516-18

Cité par Sources :