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Licensed Unlicensed Requires Authentication Published by De Gruyter July 23, 2022

A local-global theorem for p-adic supercongruences

Hao Pan, Roberto Tauraso ORCID logo and Chen Wang ORCID logo

Abstract

Let p denote the ring of all p-adic integers and call

𝒰 = { ( x 1 , , x n ) : a 1 x 1 + + a n x n + b = 0 }

a hyperplane over p n , where at least one of a 1 , , a n is not divisible by p. We prove that if a sufficiently regular n-variable function is zero modulo p r over some suitable collection of r hyperplanes, then it is zero modulo p r over the whole p n . We provide various applications of this general criterion by establishing several p-adic analogues of hypergeometric identities. For example, we confirm a conjecture of Deines et al. as follows:

k = 0 p - 1 ( 2 5 ) k 5 ( k ! ) 5 - Γ p ( 1 5 ) 5 Γ p ( 2 5 ) 5 ( mod p 5 )

for each p 1 ( mod  5 ) , where ( x ) k = x ( x + 1 ) ( x + k - 1 ) and Γ p denotes the p-adic Gamma function.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12071208

Funding statement: This work was supported by the National Natural Science Foundation of China (grant no. 12071208).

Acknowledgements

We are grateful to the anonymous referee for his/her very helpful comments on this paper.

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Received: 2021-05-17
Published Online: 2022-07-23
Published in Print: 2022-09-01

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