2020/03/24 by Chen Wang, Wang, Chen, He-Xia Ni +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT
paper · pdf · doi:10.48550/arxiv.2003.10883
7 pages
arxiv created 2020/03/24 · arxiv updated 2020/03/25
In this paper, by constructing some identities, we prove some q-analogues of some congruences. For example, for any odd integer n>1, we show that ∑k=0n-1 \frac(q-1;q2)k(q;q)k qk ≡ (-1)(n+1)/2 q(n2-1)/4 - (1+q)[n] \pmodΦn(q)2,
∑k=0n-1((q3;q2)k)/((q;q)k) qk ≡ (-1)(n+1)/2 q(n2-9)/4 + (1+q)/(q2)[n]\pmodΦn(q)2, where the q-Pochhanmmer symbol is defined by (x;q)0=1 and (x;q)k = (1-x)(1-xq)⋯(1-xqk-1) for k≥1, the q-integer is defined by [n]=1+q+⋯+qn-1 and Φn(q) is the n-th cyclotomic polynomial. The q-congruences above confirm some recent conjectures of Gu and Guo.