q-Supercongruences from Transformation Formulas

Abstract

Let $\Phi _{n}(q)$ denote the n-th cyclotomic polynomial in q. Recently, Guo and Schlosser (Constr Approx 53:155–200, 2021) put forward the following conjecture: for any odd integer $n>1$,

$$\begin{aligned}&\sum _{k=0}^{n-1}[8k-1]\frac{(q^{-1};q^4)_k^6(q^2;q^2)_{2k}}{(q^4;q^4)_k^6(q^{-1};q^2)_{2k}}q^{8k}\\&\quad \equiv {\left\{ \begin{array}{ll}0 \ (\mathrm{{mod}}\ [n]\Phi _n(q)^2), &{}\quad \text {if }n\equiv 1\ (\mathrm{{mod}}\ 4),\\ 0 \ (\mathrm{{mod}}\ [n]),&{}\quad \text {if }n\equiv 3\ (\mathrm{{mod}}\ 4). \end{array}\right. } \end{aligned}$$

where $(a;q)_k=(1-a)(1-aq)\ldots (1-aq^{k-1})$, $[n]=(1-q^n)/(1-q)$, and $\Phi _n(q)$ denotes the n-th cyclotomic polynomial in q. 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.