$q$-Supercongruences from transformation formulas

TL;DR

This paper proves a conjecture on $q$-supercongruences involving cyclotomic polynomials using advanced hypergeometric series techniques and the Chinese remainder theorem, confirming two conjectured identities for odd integers.

Contribution

It introduces the application of the 'creative microscoping' method combined with hypergeometric transformations to establish new $q$-supercongruences.

Findings

Confirmed Guo and Schlosser's conjecture for odd n.

Established a new $q$-supercongruence using hypergeometric series.

Demonstrated the effectiveness of the 'creative microscoping' method.

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.