Two congruences concerning Apéry numbers conjectured by Z.-W. Sun

Chen Wang; Chen Wang

doi:10.3934/era.2020058

Electronic Research Archive

2020, Volume 28, Issue 2: 1063-1075. doi: 10.3934/era.2020058

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Two congruences concerning Apéry numbers conjectured by Z.-W. Sun

Chen Wang

Department of Mathematics, Nanjing University, Nanjing 210093, China

Received: 01 February 2020 Revised: 01 May 2020
Primary: 11B65, 11B68; Secondary: 05A10, 11A07

Let $ n $ be a nonnegative integer. The $ n $-th Apéry number is defined by
$ A_n: = \sum\limits_{k = 0}^n\binom{n+k}{k}^2\binom{n}{k}^2. $
Z.-W. Sun investigated the congruence properties of Apéry numbers and posed some conjectures. For example, Sun conjectured that for any prime $ p\geq7 $
$ \sum\limits_{k = 0}^{p-1}(2k+1)A_k\equiv p-\frac{7}{2}p^2H_{p-1}\pmod{p^6} $
and for any prime $ p\geq5 $
$ \sum\limits_{k = 0}^{p-1}(2k+1)^3A_k\equiv p^3+4p^4H_{p-1}+\frac{6}{5}p^8B_{p-5}\pmod{p^9}, $
where $ H_n = \sum_{k = 1}^n1/k $ denotes the $ n $-th harmonic number and $ B_0, B_1, \ldots $ are the well-known Bernoulli numbers. In this paper we shall confirm these two conjectures.
- Harmonic numbers,
- Apéry numbers,
- binomial coefficients,
- congruences,
- Bernoulli numbers
Citation: Chen Wang. Two congruences concerning Apéry numbers conjectured by Z.-W. Sun[J]. Electronic Research Archive, 2020, 28(2): 1063-1075. doi: 10.3934/era.2020058

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Abstract

Let $ n $ be a nonnegative integer. The $ n $-th Apéry number is defined by

$ A_n: = \sum\limits_{k = 0}^n\binom{n+k}{k}^2\binom{n}{k}^2. $

Z.-W. Sun investigated the congruence properties of Apéry numbers and posed some conjectures. For example, Sun conjectured that for any prime $ p\geq7 $

$ \sum\limits_{k = 0}^{p-1}(2k+1)A_k\equiv p-\frac{7}{2}p^2H_{p-1}\pmod{p^6} $

and for any prime $ p\geq5 $

$ \sum\limits_{k = 0}^{p-1}(2k+1)^3A_k\equiv p^3+4p^4H_{p-1}+\frac{6}{5}p^8B_{p-5}\pmod{p^9}, $

where $ H_n = \sum_{k = 1}^n1/k $ denotes the $ n $-th harmonic number and $ B_0, B_1, \ldots $ are the well-known Bernoulli numbers. In this paper we shall confirm these two conjectures.

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