Proof of a congruence on sums of powers of q-binomial coefficients

2015/04/20 by Victor J. W. Guo, Guo, Victor J. W., Ji-Cai Liu +1
Mathematics · #05A10 #11A07 #11B65 #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:11A07 #msc:11B65

paper · pdf · doi:10.48550/arxiv.1504.05482

5 pages

arxiv created 2015/04/20 · openalex publication_date 2015/04/20 · arxiv updated 2015/04/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that, if m,n\geqslant 1 and a₁,…,a_m are nonnegative integers, then ([a₁+⋯+a_m+1]!)/([a₁]!…[a_m]!)∑^n-1_h=0q^h∏_i=1^mh\brack a_i ≡ 0\pmod[n], where [n]=(1-qⁿ)/(1-q), [n]!=[n][n-1]⋯[1], and a\brack b=∏_k=1^b\frac1-q^a-k+11-q^k. The a₁=⋯=a_m case confirms a recent conjecture of Z.-W. Sun. We also show that, if p>max\a,b\ is a prime, then ([a+b+1]!)/([a]![b]!)∑_h=0^p-1q^hh\brack ah\brack b ≡ (-1)^a-b q^ab-a\choose 2-b\choose 2[p]\pmod[p]².

Related

• Factors of sums and alternating sums of products of q-binomial coefficients and powers of q-integers
• Some q-congruences involving central q-binomial coefficients
• Binomial coefficients, roots of unity and powers of prime numbers
• Congruences involving the reciprocals of central binomial coefficients
• Super congruences concerning binomial coefficients and Apéry-like numbers