2014/08/03 by Victor J. W. Guo, Jiang Zeng, Guo, Victor J. W. +1
Mathematics · #05A10 #05A30 #11B65 #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.1408.0512
openalex publication_date 2014/08/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
For any odd prime p we obtain q-analogues of Van Hamme's supercongruence: ∑k=0(p-1)/(2)2k\choose k3(1)/(64k) ≡ 0 \pmodp2 \quadfor p≡ 3\pmod 4, and Rodriguez-Villegas' Beukers-like supercongruences involving products of three binomial coefficients. For example, we prove that ∑k=0(p-1)/(2) 2k\brack kq23 \fracq2k(-q2;q2)k2 (-q;q)2k2 amp;≡ 0\pmod[p]2 \quadfor p≡ 3\pmod 4,
∑k=0(p-1)/(2)2k\brack kq3\frac(q;q3)k (q2;q3)k q3k (q6;q6)k2 amp;≡ 0 \pmod[p]2\quadfor p≡ 2\pmod3, where [p]=1+q+⋯+qp-1, (a;q)n=(1-a)(1-aq)⋯(1-aqn-1), and n\brack kq denotes the q-binomial coefficient. Actually, our results give q-analogues of Z.-H. Sun's and Z.-W. Sun's generalizations of the above Beukers-like supercongruences. Our proof uses the theory of basic hypergeometric series including a new q-Clausen-type summation formula.