Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials

TL;DR

This paper derives new congruences involving generalized Bernoulli numbers, Bernoulli polynomials, and binomial coefficients modulo powers of integers, expanding the understanding of their number-theoretic properties.

Contribution

It introduces novel congruences for sums involving Dirichlet characters and Bernoulli polynomials, utilizing an identity by Z. H. Sun, and applies these to binomial coefficient congruences.

Findings

Established congruences for sums involving Dirichlet characters and Bernoulli polynomials.

Derived new congruences for binomial coefficients modulo n^4.

Extended known results in number theory related to Bernoulli numbers and 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 $T_{m,k}(n)=\sum_{x=1}^{[n/m]}\frac{\chi_n(x)}{x^k}\left(\bmod n^{r+1}\right)$ for $r\in \{1,2\}$, any positive integer $m $ with $n \equiv \pm 1 \left(\bmod m \right)$ 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.