Some q-supercongruences for truncated basic hypergeometric series

Victor J. W. Guo; Jiang Zeng

Acta Arithmetica, Tome 168 (2015), p. 309-326 / Harvested from The Polish Digital Mathematics Library

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 $\sum_{k = 0}^{(p - 1) / 2} {[\binom{2 k}{k}]}_{q ²}^{3} (q^{2 k}) / ((- q ²; q ²) ²_{k} (- q; q) ²_{2 k} ²) \equiv 0 (m o d [p] ²)$ for p≡ 3 (mod 4), $\sum_{k = 0}^{(p - 1) / 2} {[\binom{2 k}{k}]}_{q ³} ({(q; q ³)}_{k} {(q ²; q ³)}_{k} q^{3 k}) ({(q ⁶; q ⁶)}_{k} ²) \equiv 0 (m o d [p] ²)$ for p≡ 2 (mod 3), where $[p] = 1 + q + \dots + q^{p - 1}$ and $(a; q) ₙ = (1 - a) (1 - a q) \dots (1 - a q^{n - 1})$ . 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.

Publié le : 2015-01-01

Zbl 1338.11024

EUDML-ID : urn:eudml:doc:286367

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     title = {Some q-supercongruences for truncated basic hypergeometric series},
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     volume = {168},
     year = {2015},
     pages = {309-326},
     zbl = {1338.11024},
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Victor J. W. Guo; Jiang Zeng. Some q-supercongruences for truncated basic hypergeometric series. Acta Arithmetica, Tome 168 (2015) pp. 309-326. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa171-4-2/