q-Supercongruences from Transformation Formulas

Let $$\Phi _{n}(q)$$ denote the n-th cyclotomic polynomial in q. Recently, Guo and Schlosser (Constr Approx 53:155–200, 2021) put forward following conjecture: for any odd integer $$n>1$$ , $$\begin{aligned}&\sum _{k=0}^{n-1}[8k-1]\frac{(q^{-1};q^4)_k^6(q^2;q^2)_{2k}}{(q^4;q^4)_k^6(q^{-1};q^2)_{2k}}q^{8k}\\&\quad \equiv {\left\{ \begin{array}{ll}0 \ (\mathrm{{mod}}\ [n]\Phi _n(q)^2), &{}\quad \text {if }n\equiv 1\ 4),\\ 0 [n]),&{}\quad 3\ 4). \end{array}\right. } \end{aligned}$$ where $$(a;q)_k=(1-a)(1-aq)\ldots (1-aq^{k-1})$$ $$[n]=(1-q^n)/(1-q)$$ _n(q)$$ denotes Applying ‘creative microscoping’ method several summation transformation formulas basic hypergeometric series Chinese remainder theorem coprime polynomials, we confirm above conjecture, as well another similar q-supercongruence conjectured by Schlosser.