Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 71, No. 4, pp. 1157-1165, 2021

A $q$-congruence for a truncated $_4\varphi_3$ series

Victor J. W. Guo, Chuanan Wei

Received July 26, 2020. Published online June 18, 2021.

Abstract: 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 $n\equiv1\pmod4$,
\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} \equiv0\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 $n\equiv3\pmod4$. Our proof is based on the `creative microscoping' method, recently developed by Guo and Zudilin, and a $_4\varphi_3$ summation formula.

Keywords: basic hypergeometric series; Watson's transformation; $q$-congruence; supercongruence; creative microscoping

Classification MSC: 33D15, 11A07, 11B65

DOI: 10.21136/CMJ.2021.0317-20

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

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Affiliations: Victor J. W. Guo, School of Mathematics and Statistics, Huaiyin Normal University, 111 West Changjiang Road, Huai'an 223300, Jiangsu, P. R. China, e-mail: jwguo@hytc.edu.cn; Chuanan Wei (corresponding author), School of Biomedical Information and Engineering, Hainan Medical University, No 3, Xueyuan Road, Haikou 571199, P. R. China, e-mail: weichuanan78@163.com

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