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Some q-supercongruences for truncated basic hypergeometric series

100%

Guo V., Zeng J.

Acta Arithmetica

2015

tom 171

nr 4

309-326

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.

Factors of alternating sums of products of binomial and q-binomial coefficients

81%

Guo V., Jouhet F., Zeng J.

Acta Arithmetica

2007

tom 127

nr 1

17-31

Ograniczanie wyników

2 Acta Arithmetica

2 Guo V.

2 Zeng J.

1 Jouhet F.

1 2015

1 2007

Wyniki wyszukiwania

Some q-supercongruences for truncated basic hypergeometric series

Factors of alternating sums of products of binomial and q-binomial coefficients