On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ₁F₀ series

2018/10/22 by Zhang, Yong, Pan, Hao
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1810.09370

Abstract

Let p be an odd prime and let n be a positive integer. For any positive integer α and m∈\1,2,3\, we have ∑_k=0^pαn-1((\frac12)_k)/(k!)⋅((-4)^k)/(m^k)≡((m(m-4))/(p))∑_k=0^p^α-1n-1((\frac12)_k)/(k!)⋅((-4)^k)/(m^k)\pmodp^2α, where (x)_k=x(x+1)⋯(x+k-1) and ((⋅)/(⋅)) denotes the Legendre symbol. Also, when m=4, ∑_k=0^pαn-1(-1)^k⋅((\frac12)_k)/(k!)≡ p∑_k=0^p^α-1n-1(-1)^k⋅((\frac12)_k)/(k!)\pmodp^2α.

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