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arXiv · 2211.15874

Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials

Ni Li·Rong Ma
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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.

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Ni Li, Rong Ma. 2022-11-29. Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials. https://arxiv.org/abs/2211.15874

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