2020/02/20 by Chen Wang, Wei Xia, Wang, Chen +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2002.08814
openalex publication_date 2020/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, using the well-known Karlsson-Minton formula, we mainly establish two divisibility results concerning truncated hypergeometric series. Let n>2 and q>0 be integers with 2| n or 2\nmid q. We show that ∑k=0p-1((q-(p)/(n))kn)/((1)kn)≡0\pmodp3 and pn∑k=0p-1((1)kn)/(((p)/(n)-q+2)kn)≡0\pmodp3 for any prime p>max\n,(q-1)n+1\, where (x)k denotes the Pochhammer symbol defined by (x)k=\begincases1, amp;k=0,
x(x+1)⋯(x+k-1), amp;kgt;0.\endcases Let n≥4 be an even integer. Then for any prime p with p≡-1\pmodn, the first congruence above implies that ∑k=0p-1 (((1)/(n))kn)/((1)kn)≡0\pmodp3. This confirms a recent conjecture of Guo.