Some q-congruences related to 3-adic valuations

2009/10/22 by Hao Pan, Zhi-Wei Sun, Zhi‐Wei Sun +2 · 1 citation
Mathematics · #05A10 #05A30 #11A07 #11B65 #11S99 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A10 #msc:05A30 #msc:11A07 #msc:11B65 #msc:11S99

paper · pdf · doi:10.48550/arxiv.0910.4170

9 pages

openalex publication_date 2009/10/22 · arxiv created 2009/10/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1992 Strauss, Shallit and Zagier proved that for any positive integer a we have ∑_k=0^3a-1\binom2kk=0 (mod 3^2a) and furthermore 3^-2a∑_k=0^3a-1\binom2kk=1 (mod 3). Recently a q-analogue of the former congruence was conjectured by Guo and Zeng. In this paper we prove the Guo-Zeng conjecture and also give a q-analogue of the latter congruence.

Citations

• Various congruences involving binomial coefficients and higher-order Catalan numbers

Cited by

• q-Congruences for Z.-W. Sun's generalized polynomials w^(α)_k(x)

Related

• The sum ∑_k=0^q-1\binom2kk for q a power of 3
• Congruences for sums involving products of three binomial coefficients
• Some 3-adic congruences for binomial sums
• Cubic congruences and sums involving \binom3kk
• Congruences involving \binom2kk²\binom3kkm^-k