A q-ANALOGUE OF A HYPERGEOMETRIC CONGRUENCE

被引：6
作者：
Gu, Cheng-Yang [1 ]
Guo, Victor J. W. [1 ]
机构：
[1] Huaiyin Normal Univ, Sch Math Sci, Huaian 223300, Jiangsu, Peoples R China
来源：
BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY | 2020年 / 101卷 / 02期
基金：
中国国家自然科学基金;
关键词：
basic hypergeometric series; q-congruence; congruence; cyclotomic polynomia; SUPERCONGRUENCE; FORMULAS;
D O I：
10.1017/S0004972719000777
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
We give a q-analogue of the following congruence: for any odd prime p, \sum_{k=0}<^>{(p-1)/2}(-1)<^>k(6k+1)\frac{(\frac{1}{2})_k<^>3}{k!<^>3 8<^>k}\sum_{j=1}<^>{k}(\frac{1}{(2j-1)<^>2}-\frac{1}{16j<^>2}) \equiv 0\pmod{p}, which was originally conjectured by Long and later proved by Swisher. This confirms a conjecture of the second author ['A q-analogue of the (L.2) supercongruence of Van Hamme', J. Math. Anal. Appl. 466 (2018), 749761].
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页码：294 / 298
页数：5
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