New congruences involving harmonic numbers

Zhi-Wei Sun

Published 2014-07-31, updated 2024-01-09Version 5

Let $p>3$ be a prime. For any $p$-adic integer $a$, we determine $$\sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}kH_k,\ \ \sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}kH_k^{(2)},\ \ \sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}k\frac{H_k^{(2)}}{2k+1}$$ modulo $p^2$, where $H_k=\sum_{0<j\le k}1/j$ and $H_k^{(2)}=\sum_{0<j\le k}1/j^2$. In particular, we show that \begin{gather*}\sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}kH_k\equiv(-1)^{\langle a\rangle_p}\,2\left(B_{p-1}(a)-B_{p-1}\right)\pmod p, \\\sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}kH_k^{(2)}\equiv -E_{p-3}(a)\pmod p, \\(2a-1)\sum_{k=0}^{p-1}\binom{-a}k\binom{a-1}k\frac{H_k^{(2)}}{2k+1}\equiv B_{p-2}(a)\pmod p, \end{gather*} where $\langle a\rangle_p$ stands for the least nonnegative integer $r$ with $a\equiv r\pmod{p}$, and $B_n(x)$ and $E_n(x)$ denote the Bernoulli polynomial of degree $n$ and the Euler polynomial of degree $n$ respectively. We also pose some new conjectures on congruences.