摘要
设p>3为素数.对任何p-adic整数a,我们决定出p−1 X k=0 −k a a−1 k Hk,p−1 X k=0 −k a a−1 k Hk(2),p−1 X k=0 −k a a−1 k H(2)k 2k+1模p 2,其中Hk=P 0<j<k 1/j且Hk(2)=P 0<j>k 1/j2.特别地,我们证明了p−1 X k=0 −k a a−1 k Hk≡(−1)p 2(Bp−1(a)−Bp−1)(mod p),p−1 X k=0 −k a a−1 k Hk(2)≡−Ep−3(a)(mod p),(2a−1)p−1 X k=0 −k a a−1 k H(2)k 2k+1≡Bp−2(a)(mod p)其中p表示满足a≤r(mod p)的最小非负整数r,Bn(x)与En(x)分别表示次数为n的伯努利多项式与欧拉多项式.
Let p>3 beaprime.Forany p-adic integer a,wedetermine p−1 X k=0−k aa−1 kHk,p−1 X k=0−k aa−1 kHk(2),p−1 X k=0−k aa−1 kH(2)k 2k+1 modulo p2,where Hk=P 0<j<k 1/j and Hk(2)=P 0<j>k 1/j2.In particular,we show that p−1 X k=0−k aa−1 kHk≡(−1)p 2(Bp−1(a)−Bp−1)(mod p),p−1 X k=0−k aa−1 kHk(2)≡−Ep−3(a)(mod p),(2a−1)p−1 X k=0−k aa−1 kH(2)k 2k+1≡Bp−2(a)(mod p),wherep stands for the least nonnegative integer r with a≡r(mod p),and Bn(x)and En(x)denote the Bernoulli polynomial of degree n and the Euler polynomial of degree n respectively.We also pose some new conjectures on congruences.
作者
孙智伟
Sun Zhiwei(Department of Mathematics,Nanjing University,Nanjing 210093)
出处
《南京大学学报（数学半年刊）》
2023年第1期1-32,共32页
Journal of Nanjing University(Mathematical Biquarterly)
基金
Supported by the National Natural Science Foundation of China(11971222)
the initial version was posted to arXiv in 2014 with the ID arXiv:1407.8465.
关键词
伯努利与欧拉多项式
二项式系数
同余式
调和数
Bernoulli and Euler polynomials
binomial coefficients
congruences
harmonic numbers
