Some $q$-supercongruences for truncated basic hypergeometric series

Victor J. W. Guo ¹   ; Jiang Zeng ²

¹ School of Mathematical Sciences Huaiyin Normal University Huaian, Jiangsu 223300 People's Republic of China
² Université de Lyon Université Lyon 1 Institut Camille Jordan, UMR 5208 du CNRS 43, boulevard du 11 novembre 1918 F-69622 Villeurbanne Cedex, France

Acta Arithmetica, Tome 171 (2015) no. 4, pp. 309-326

Résumé

For any odd prime $p$ we obtain $q$-analogues of van Hamme's and Rodriguez-Villegas' supercongruences involving products of three binomial coefficients such as \begin{align*} \sum_{k=0}^{{(p-1)}/{2}} \bigg[{2k\atop k}\bigg]_{q^2}^3 \frac{q^{2k}}{(-q^2;q^2)_k^2 (-q;q)_{2k}^2} \equiv 0 \pmod{[p]^2} \ \text{for}\ p\equiv 3 \pmod 4, \\ \sum_{k=0}^{{(p-1)}/{2}}\bigg[{2k\atop k}\bigg]_{q^3}\frac{(q;q^3)_k (q^{2};q^3)_{k} q^{3k} }{ (q^{6};q^{6})_k^2 } \equiv 0 \pmod{[p]^2}\ \text{for}\ p\equiv 2 \pmod{3}, \end{align*} where $[p]=1+q+\cdots+q^{p-1}$ and $(a;q)_n=(1-a)(1-aq)\cdots(1-aq^{n-1})$. We also prove $q$-analogues of the Sun brothers' generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial $q$-binomial identities including a new $q$-Clausen type summation formula.

Détail

DOI : 10.4064/aa171-4-2

Keywords: odd prime obtain q analogues van hammes rodriguez villegas supercongruences involving products three binomial coefficients begin align* sum ffgrac p bigg atop bigg frac q q equiv pmod text equiv pmod sum ffgrac p bigg atop bigg frac equiv pmod text equiv pmod end align* where cdots p a aq cdots aq n prove q analogues sun brothers generalizations above supercongruences proofs elementary nature theory basic hypergeometric series combinatorial q binomial identities including q clausen type summation formula