2021/03/04 by Chen Wang, Zhi‐Wei Sun, Wang, Chen +1
Mathematics · #05A10 #11A07 #11B65 #33C20 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2103.02951
openalex publication_date 2021/03/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For any m,n∈ℕ=\0,1,2…\, the truncated hypergeometric series m+1Fm is defined by m+1Fm[\beginmatrixx0amp;x1amp;\ldotsamp;xm
amp;y1amp;\ldotsamp;ym\endmatrix|z]n=∑k=0n((x0)k(x1)k⋯(xm)k)/((y1)k⋯(ym)k)⋅(zk)/(k!), where (x)k=x(x+1)⋯(x+k-1) is the Pochhammer symbol. Let p be an odd prime. For α,z∈ℤp with ⟨ -α⟩p≡0\pmod2, where ⟨ x⟩p denotes the least nonnegative residue of x modulo p for any x∈ℤp, we mainly prove the following congruence motivated by Orr's identity: 2F1[\beginmatrix\frac12αamp;\frac32-\frac12α
amp;1\endmatrix|z]p-12F1[\beginmatrix\frac12αamp;\frac12-\frac12α
amp;1\endmatrix|z]p-1≡3F2[\beginmatrixαamp;2-αamp;\frac12
amp;1amp;1\endmatrix|z]p-1\pmodp2. As a corollary, for any positive integer b with p≡±1\pmodb and ⟨ -1/b⟩p≡0\pmod2, we deduce that ∑k=0p-1(b2k+b-1)\frac\binom2kk4k\binom-1/bk\binom1/b-1k≡0\pmodp2. This confirms a conjectural congruence of the second author.