Two congruences involving harmonic numbers with applications

Abstract

The harmonic numbers $H_n={\sum }_{0<k\le n}1/k(n=0,1,2,\dots )$ play important roles in mathematics. Let $p>3$ be a prime. With helps of some combinatorial identities, we establish the following two new congruences: $$\sum _{k=1}^{p-1}\frac{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{k}{H}_k\equiv \frac{1}{3}\left(\frac{p}{3}\right)B_{p-2}\left(\frac{1}{3}\right)(\mathrm{mod}p)$$ and $$\sum _{k=1}^{p-1}\frac{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}{k}{H}_{2k}\equiv \frac{7}{12}\left(\frac{p}{3}\right)B_{p-2}\left(\frac{1}{3}\right)(\mathrm{mod}p),$$ where $B_n(x)$ denotes the Bernoulli polynomial of degree $n$. As an application, we determine ${\sum }_{n=1}^{p-1}{g}_n$ and ${\sum }_{n=1}^{p-1}{h}_n$ modulo $p^3$, where $$g_n=\sum_{k=0}^n\binom nk^2\binom{2k}k\quad\mbox{and}\quad h_n=\sum_{k=0}^n\binom nk^2C_k$$ with $C_k=\left(\genfrac{}{}{0ex}{}{2k}{k}\right)/(k+1)$.

14 pages

References in corpus (3)

• New series for some special values of $L$-functions
• On sums of binomial coefficients modulo p^2
• Congruences involving $g_n(x)={\sum }_{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{2k}{k}\right)x^k$