# New congruences involving harmonic numbers

Research paper by **Zhi-Wei Sun**

Indexed on: **11 Aug '14**Published on: **11 Aug '14**Published in: **Mathematics - Number Theory**

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#### Abstract

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.