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Supercongruences concerning truncated hypergeometric series

2018/06/07 by Wang, Chen, Pan, Hao
#05A10 #11A07 #11B65 #33C20 (Primary) #33E50 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1806.02735

Abstract

Let n≥ 3 be an integer and p be a prime with p≡ 1\pmodn. In this paper, we show that nFn-1[\beginmatrix (n-1)/(n)amp;(n-1)/(n)amp;\ldotsamp;(n-1)/(n)
amp;1amp;\ldotsamp;1\endmatrix | 1]p-1≡ -Γp((1)/(n))n\pmodp3, where the truncated hypergeometric series nFn-1 [\beginmatrix x1amp;x2amp;\ldotsamp;xn
amp;y1amp;\cdotsamp;yn-1\endmatrix | z]m=∑k=0m(zk)/(k!)∏j=0k-1\frac(x1+j)⋯(xn+j)(y1+j)⋯(yn-1+j) and Γp denotes the p-adic gamma function. This confirms a conjecture of Deines, Fuselier, Long, Swisher and Tu. Furthermore, under the same assumptions, we also prove that pn⋅ n+1 Fn [ \beginmatrix 1 amp;1 amp;… amp;1
amp;(n+1)/(n) amp;… amp;(n+1)/(n) \endmatrix | 1]p-1 ≡ -Γp ((1)/(n) )n (mod p3), which solves another conjecture of Deines, Fuselier, Long, Swisher and Tu.

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