2023/10/23 by Victor J. W. Guo, Guo, Victor J. W.
Mathematics · Medicine · #11A07 #11B65 #Advanced Mathematical Identities #FOS: Mathematics #Leprosy Research and Treatment #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2310.15207
openalex publication_date 2023/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Employing the q-Lucas theorem and some known q-supercongruences, we give some Dwork-type q-congruences, confirming three conjectures in [J. Combin. Theory, Ser. A 178 (2021), Art.~105362]. As conclusions, we obtain the following supercongruences: for any prime p≡ 1\pmod4 and positive integer r, ∑k=0(pr-1)/2 (((1)/(2))k3)/(k!3) amp;≡ -Γp(\tfrac14)4 ∑k=0^(pr-1-1)/2 (((1)/(2))k3)/(k!3) \pmodpr+1,
∑k=0pr-1 (((1)/(2))k3)/(k!3) amp;≡ -Γp(\tfrac14)4 ∑k=0^pr-1-1 (((1)/(2))k3)/(k!3) \pmodpr+1, where Γp(x) stands for the p-adic Gamma function. The first one confirms a weaker form of Swisher's (H.3) conjecture for p≡ 1\pmod4, which originally predicts that the supercongruence is true modulo p3r.