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A $q$-ANALOGUE OF A HYPERGEOMETRIC CONGRUENCE

Part of: Elementary number theory Discontinuous groups and automorphic forms Basic hypergeometric functions

Published online by Cambridge University Press:  18 July 2019

CHENG-YANG GU Open the ORCID record for CHENG-YANG GU [Opens in a new window]  and
VICTOR J. W. GUO Open the ORCID record for VICTOR J. W. GUO [Opens in a new window]
CHENG-YANG GU
Affiliation:
School of Mathematical Sciences, Huaiyin Normal University, Huai’an 223300, Jiangsu, PR China email 525290408@qq.com
VICTOR J. W. GUO*
Affiliation:
School of Mathematical Sciences, Huaiyin Normal University, Huai’an 223300, Jiangsu, PR China email jwguo@math.ecnu.edu.cn
*
✉jwguo@math.ecnu.edu.cn
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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].

Keywords

basic hypergeometric seriesq-congruencecongruencecyclotomic polynomial

MSC classification

Primary: 33D15: Basic hypergeometric functions in one variable, $_rphi_s$
Secondary: 11A07: Congruences; primitive roots; residue systems 11F33: Congruences for modular and $p$-adic modular forms
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 2 , April 2020 , pp. 294 - 298
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

The second author was partially supported by the National Natural Science Foundation of China (grant 11771175).

References

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