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Some $q$-congruences with parameters

Volume 190 / 2019

Victor J. W. Guo Acta Arithmetica 190 (2019), 381-393 MSC: 33D15, 11A07, 11B65. DOI: 10.4064/aa180624-16-12 Published online: 29 July 2019


Let $\varPhi_n(q)$ be the $n$th cyclotomic polynomial in $q$. Recently, the author and Zudilin devised a method, called ‘creative microscoping’, to prove some $q$-supercongruences mainly modulo $\varPhi_n(q)^3$ by introducing an additional parameter $a$. In this paper, we use this method to confirm some conjectures on $q$-supercongruences modulo $\varPhi_n(q)^2$. We also give some parameter-generalizations of known $q$-supercongruences. For instance, we present further generalizations of a $q$-analogue of a famous supercongruence of Rodriguez-Villegas:


  • Victor J. W. GuoSchool of Mathematical Sciences
    Huaiyin Normal University
    Huai’an 223300, Jiangsu, People’s Republic of China

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