Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials

Abstract

Let $[x]$ be the integral part of $x$, $n>1$ be a positive integer and ${\chi }_n$ denote the trivial Dirichlet character modulo $n$. In this paper, we use an identity established by Z. H. Sun to get congruences of $T_{m,k}(n)={\sum }_{x=1}^{[n/m]}\frac{{\chi }_n(x)}{x^k}\left(\mathrm{mod}{n}^{r+1}\right)$ for $r\in \{1,2\}$, any positive integer $m$ with $n\equiv \pm 1\left(\mathrm{mod}m\right)$ in terms of Bernoulli polynomials. As its an application, we also obtain some new congruences involving binomial coefficients modulo $n^4$ in terms of generalized Bernoulli numbers.

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