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Keywords:
basic hypergeometric series; Watson's transformation; $q$-congruence; supercongruence; creative microscoping
Summary:
Let $\Phi _n(q)$ denote the $n$th cyclotomic polynomial in $q$. Recently, Guo, Schlosser and Zudilin proved that for any integer $n>1$ with $n\equiv 1\pmod {4}$, $$ \sum _{k=0}^{n-1}\frac {(q^{-1};q^2)_k^2(q^{-2};q^4)_k}{(q^2;q^2)_k^2 (q^4;q^4)_k}q^{6k} \equiv 0\pmod {\Phi _n(q)^2}, $$ where $(a;q)_m=(1-a)(1-aq)\cdots (1-aq^{m-1})$. In this note, we give a generalization of the above $q$-congruence to the modulus $\Phi _n(q)^3$ case. Meanwhile, we give a corresponding $q$-congruence modulo $\Phi _n(q)^2$ for $n\equiv 3\pmod {4}$. Our proof is based on the `creative microscoping' method, recently developed by Guo and Zudilin, and a $_4\varphi _3$ summation formula.
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