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arXiv · 2109.12034

$q$-Supercongruences from transformation formulas

He-Xia Ni·Li-Yuan Wang·Hai-Liang Wu
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Abstract

Let $\Phi_{n}(q)$ denote the $n$-th cyclotomic polynomial in $q$. Recently, Guo and Schlosser [Constr. Approx. 53 (2021), 155--200] put forward the following conjecture: for an odd integer $n>1$, \begin{align*} &\sum_{k=0}^{n-1}[8k-1]\frac{(q^{-1};q^4)_k^6(q^2;q^2)_{2k}}{(q^4;q^4)_k^6(q^{-1};q^2)_{2k}}q^{8k}\notag\\ &\quad\equiv\begin{cases}0 \pmod{[n]\Phi_n(q)^2}, &\text{if }n\equiv 1\pmod{4},\\[5pt] 0 \pmod{[n]},&\text{if }n\equiv 3\pmod{4}. \end{cases} \end{align*} Applying the `creative microscoping' method and several summation and transformation formulas for basic hypergeometric series and the Chinese remainder theorem for coprime polynomials, we confirm the above conjecture, as well as another similar $q$-supercongruence conjectured by Guo and Schlosser.

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He-Xia Ni, Li-Yuan Wang, Hai-Liang Wu. 2021-09-24. $q$-Supercongruences from transformation formulas. https://arxiv.org/abs/2109.12034

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