Some q-supercongruences for truncated basic hypergeometric series

Victor J. W. Guo; Jiang Zeng

Acta Arithmetica (2015)

  • Volume: 171, Issue: 4, page 309-326
  • ISSN: 0065-1036

Abstract

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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 [ 2 k k ] q ² 3 ( q 2 k ) / ( ( - q ² ; q ² ) ² k ( - q ; q ) ² 2 k ² ) ≡ 0 ( m o d [ p ] ² ) for p≡ 3 (mod 4), ∑ k = 0 ( p - 1 ) / 2 [ 2 k k ] q ³ ( ( q ; q ³ ) k ( q ² ; q ³ ) k q 3 k ) ( ( q ⁶ ; q ⁶ ) k ² ) ≡ 0 ( m o d [ p ] ² ) for p≡ 2 (mod 3), where [ p ] = 1 + q + ⋯ + q p - 1 and ( a ; q ) ₙ = ( 1 - a ) ( 1 - a q ) ⋯ ( 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.

How to cite

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Victor J. W. Guo, and Jiang Zeng. "Some q-supercongruences for truncated basic hypergeometric series." Acta Arithmetica 171.4 (2015): 309-326. <http://eudml.org/doc/286367>.

@article{VictorJ2015,
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 $∑_\{k=0\}^\{(p-1)/2\} [\{2k \atop k\}]_\{q²\}^\{3\} (q^\{2k\})/((-q²;q²)²_\{k\}(-q;q)²_\{2k\}²) ≡ 0 (mod [p]²)$ 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.},
author = {Victor J. W. Guo, Jiang Zeng},
journal = {Acta Arithmetica},
keywords = {-adic integer; congruences; supercongruences; basic hypergeometric series; q-binomial theorem; -Chu-Vandermonde formula},
language = {eng},
number = {4},
pages = {309-326},
title = {Some q-supercongruences for truncated basic hypergeometric series},
url = {http://eudml.org/doc/286367},
volume = {171},
year = {2015},
}

TY - JOUR
AU - Victor J. W. Guo
AU - Jiang Zeng
TI - Some q-supercongruences for truncated basic hypergeometric series
JO - Acta Arithmetica
PY - 2015
VL - 171
IS - 4
SP - 309
EP - 326
AB - 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.
LA - eng
KW - -adic integer; congruences; supercongruences; basic hypergeometric series; q-binomial theorem; -Chu-Vandermonde formula
UR - http://eudml.org/doc/286367
ER -

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