Some q-supercongruences for truncated basic hypergeometric series

• Autorzy: Victor J. W. Guo , Jiang Zeng
• Języki publikacji: EN

Abstrakty

EN

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
$∑_{k=0}^{(p-1)/2} [{2k \atop k}]_{q²}^{3} (q^{2k})/((-q²;q²)²_{k}(-q;q)²_{2k}²) ≡ 0 (mod [p]²)$ for p≡ 3 (mod 4),
$∑_{k=0}^{(p-1)/2} [{2k \atop k}]_{q³} ((q;q³)_{k}(q²;q³)_{k}q^{3k})((q⁶;q⁶)_{k}²) ≡ 0 (mod [p]²)$ for p≡ 2 (mod 3),
where $[p] = 1 + q + ⋯ +q^{p-1}$ and $(a;q)ₙ = (1-a)(1-aq)⋯ (1-aq^{n-1})$. 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.

Kategorie tematyczne

• 05A30: q -calculus and related topics
• 11B65: Binomial coefficients; factorials; q -identities
• 05A10: Factorials, binomial coefficients, combinatorial functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 171
• Numer: 4
• Strony: 309-326

Daty

wydano

2015

Twórcy

autor

Victor J. W. Guo

• School of Mathematical Sciences, Huaiyin Normal University, Huaian, Jiangsu 223300, People's Republic of China

autor

Jiang Zeng

• Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR 5208 du CNRS, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France

Identyfikatory

DOI

10.4064/aa171-4-2