New congruences involving harmonic numbers

Abstract

Let $p>3$ be a prime. For any $p$-adic integer $a$, we determine $$\sum _{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{-a}{k}\right)\left(\genfrac{}{}{0ex}{}{a-1}{k}\right)H_k,\sum _{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{-a}{k}\right)\left(\genfrac{}{}{0ex}{}{a-1}{k}\right)H_k^{(2)},\sum _{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{-a}{k}\right)\left(\genfrac{}{}{0ex}{}{a-1}{k}\right)\frac{H_k^{(2)}}{2k+1}$$ 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 $⟨a{⟩}_p$ stands for the least nonnegative integer $r$ with $a\equiv r(\mathrm{mod}p)$, 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.

32 pages, final published version

References in corpus (2)

• Some q-analogues of supercongruences of Rodriguez-Villegas
• Super congruences involving Bernoulli and Euler polynomials

Cited by in corpus (1)

• Supercongruences related to ${}_3{F}_2(1)$ involving harmonic numbers