A+ CATEGORY SCIENTIFIC UNIT

Some $q$-supercongruences for truncated basic hypergeometric series

Volume 171 / 2015

Victor J. W. Guo, Jiang Zeng Acta Arithmetica 171 (2015), 309-326 MSC: Primary 11B65; Secondary 05A10, 05A30. DOI: 10.4064/aa171-4-2

Abstract

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.

Authors

  • Victor J. W. GuoSchool of Mathematical Sciences
    Huaiyin Normal University
    Huaian, Jiangsu 223300
    People's Republic of China
    e-mail
  • Jiang ZengUniversité de Lyon
    Université Lyon 1
    Institut Camille Jordan, UMR 5208 du CNRS
    43, boulevard du 11 novembre 1918
    F-69622 Villeurbanne Cedex, France
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image