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A $\boldsymbol {q}$-SUPERCONGRUENCE MODULO THE THIRD POWER OF A CYCLOTOMIC POLYNOMIAL

Part of: Elementary number theory Basic hypergeometric functions Sequences and sets

Published online by Cambridge University Press:  18 July 2022

CHUANAN WEI Open the ORCID record for CHUANAN WEI [Opens in a new window]
CHUANAN WEI*
Affiliation:
School of Biomedical Information and Engineering, Hainan Medical University, Haikou 571199, PR China
*
e-mail: weichuanan78@163.com
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Abstract

We derive a q-supercongruence modulo the third power of a cyclotomic polynomial with the help of Guo and Zudilin’s method of creative microscoping [‘A q-microscope for supercongruences’, Adv. Math. 346 (2019), 329–358] and the q-Dixon formula. As consequences, we give several supercongruences including

$$ \begin{align*}\sum_{k=0}^{(p-2)/3}\frac{(\frac{2}{3})_k^3}{(1)_k^3}\equiv\frac{p}{2}\frac{(1)_{(p-2)/3}}{(\frac{4}{3})_{(p-2)/3}}\pmod{p^3},\end{align*} $$

where p is a prime with $p\equiv 5\pmod {6}$ .

Keywords

basic hypergeometric seriesq-Dixon formulaq-supercongruence

MSC classification

Primary: 11B65: Binomial coefficients; factorials; $q$-identities
Secondary: 11A07: Congruences; primitive roots; residue systems 33D15: Basic hypergeometric functions in one variable, $_rphi_s$
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 236 - 242
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This work is supported by the National Natural Science Foundation of China (No. 12071103).

References

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