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A 
$\boldsymbol {q}$-SUPERCONGRUENCE MODULO THE THIRD POWER OF A CYCLOTOMIC POLYNOMIAL
Published online by Cambridge University Press: 18 July 2022
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
$p\equiv 5\pmod {6}$
.
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 236 - 242
- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
This work is supported by the National Natural Science Foundation of China (No. 12071103).
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