A $q$-ANALOGUE OF A HYPERGEOMETRIC CONGRUENCE. Issue 2 (18th April 2020)

• Record Type: Journal Article
• Authors: GU, CHENG-YANG; GUO, VICTOR J. W.
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 101:Issue 2(2020)
• Page Start: 294
• Page End: 298
• Publication Date: 2020-04-18
• Subjects: 33D15, -- 11A07, -- 11F33; basic hypergeometric series, -- q-congruence, -- congruence, -- cyclotomic polynomial; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972719000777
• Languages: English
• ISSNs: 0004-9727

Abstract

Abstract : We give a $q$ -analogue of the following congruence: for any odd prime $p$, $$\begin{eqnarray}\mathop{\sum }_{k=0}^{(p-1)/2}(-1)^{k}(6k+1)\frac{(\frac{1}{2})_{k}^{3}}{k!^{3}8^{k}}\mathop{\sum }_{j=1}^{k}\biggl(\frac{1}{(2j-1)^{2}}-\frac{1}{16j^{2}}\biggr)\equiv 0\;(\text{mod}\;p), \end{eqnarray}$$ 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), 749–761].