Some q-congruences with parameters

2018/04/29 by Guo, Victor J. W.
#11B65 #11F33 #33D15 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1804.10963

Abstract

Let Φ_n(q) be the n-th cyclotomic polynomial in q. Recently, the author and Zudilin provide a creative microscoping method to prove some q-supercongruences mainly modulo Φ_n(q)³ by introducing an additional parameter a. In this paper, we use this creative microscoping method to confirm some conjectures on q-supercongruences modulo Φ_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: ∑_k=0^p-1\frac2k\choose k²16^k ≡ (-1)^(p-1)/2\pmodp²\quadfor any odd prime p.

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