Some q-analogues of supercongruences of Rodriguez-Villegas

JOURNAL OF NUMBER THEORY

卷:145

Article

Guo, Victor J. W.¹  Zeng, Jiang² 

[1] E China Normal Univ, Shanghai Key Lab PMMP, Dept Math, Shanghai 200241, Peoples R China

[2] Univ Lyon 1, Inst Camille Jordan, CNRS, UMR 5208, F-69622 Villeurbanne, France

关键词: Congruences;    Least nonnegative residue;    Little q-Legendre polynomials;    q-Binomial theorem;    q-Chu-Vandermonde formula;   

DOI  :  10.1016/j.jnt.2014.06.002

来源: Elsevier

【 摘 要 】

We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence [GRAPHICS] , where (($p) over bar) is the Legendre symbol, has the following two nice q-analogues: [GRAPHICS] equivalent to (-1/p)q((p2-1)e/4) (mod (1 + q + ... + q(p-1))(2)), where (a; q) = (1 - a)(1 - aq) ... (1 - aq(n-1)) and epsilon = +/- 1. Several related conjectures are also proposed. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

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