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Some q-supercongruences for truncated basic hypergeometric series

Victor J. W. GuoJiang Zeng — 2015

Acta Arithmetica

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

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