On two congruences involving Franel numbers

2020/02/10 by Ji-Cai Liu, Liu, Ji-Cai
Mathematics · #05A19 #11A07 #33F10 #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.2002.03650

openalex publication_date 2020/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Via symbolic summation method, we establish the following series for π²: ∑_k=1^∞ \fracH_k-2H_2k(-3)^k k = (π²)/(18), where H_k=∑_j=1^k 1/j. We also derive a p-adic congruence related to this series. As an application, we prove two congruences involving Franel numbers, one of which was originally conjectured by Sun.

Citations

• List of conjectural series for powers of π and other constants
• A q-microscope for supercongruences
• Infinite Summation Formulas Involving Riemann-Zeta function

Related

• Proof of a supercongruence conjectured by Z.-H. Sun
• Congruences involving Franel and Catalan-Larcombe-French numbers
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