q-Analogues of some supercongruences related to Euler numbers

2020/10/20 by Victor J. W. Guo, Guo, Victor J. W. · 1 citation
Mathematics · #11A07 #11B65 #33F10 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2010.13526

openalex publication_date 2020/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let E_n be the n-th Euler number and (a)_n=a(a+1)⋯ (a+n-1) the rising factorial. Let p>3 be a prime. In 2012, Sun proved the that ∑^(p-1)/2_k=0(-1)^k(4k+1)(((1)/(2))_k³)/(k!³) ≡ p(-1)^(p-1)/2+p³E_p-3 \pmodp⁴, which is a refinement of a famous supercongruence of Van Hamme. In 2016, Chen, Xie, and He established the following result: ∑_k=0^p-1(-1)^k (3k+1)(((1)/(2))_k³)/(k!³) 2^3k ≡ p(-1)^(p-1)/2+p³E_p-3 \pmodp⁴, which was originally conjectured by Sun. In this paper we give q-analogues of the above two supercongruences by employing the q-WZ method. As a conclusion, we provide a q-analogue of the following supercongruence of Sun: ∑_k=0^(p-1)/2(((1)/(2))_k²)/(k!²) ≡ (-1)^(p-1)/2+p² E_p-3 \pmodp³.

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• A refined q-analogue of some congruences of Van Hamme

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