A p-adic analogue of Chan and Verrill's formula for 1/π

2020/08/15 by Ji-Cai Liu, Liu, Ji-Cai
Mathematics · #05A19 #11A07 #11B65 #11Y55 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2008.06675

openalex publication_date 2020/08/15 · openalex created_date 2020/08/21 · openalex updated_date 2026/07/28

Abstract

We prove three supercongruences for sums of Almkvist-Zudilin numbers, which confirm some conjectures of Zudilin and Z.-H. Sun. A typical example is the Ramanujan-type supercongruence: ∑_k=0^p-1 (4k+1)/(81^k)γ_k ≡ ((-3)/(p)) p\pmodp³, which is corresponding to Chan and Verrill's formula for 1/π: ∑_k=0^∞ (4k+1)/(81^k)γ_k = (3√(3))/(2π). Here γ_n are the Almkvist-Zudilin numbers.

Citations

• A q-microscope for supercongruences

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