Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials
Tom 212 / 2024
Acta Arithmetica 212 (2024), 49-69
MSC: Primary 11B68; Secondary 11A07
DOI: 10.4064/aa221130-8-8
Opublikowany online: 8 January 2024
Streszczenie
Let be the integral part of x, n \gt 1 be a positive integer and \chi_n denote the trivial Dirichlet character modulo n. We use an identity established by Z. H. Sun to get congruences of T_{m,k}(n)=\sum_{i=1}^{[n/m]}\frac{\chi_n(i)}{i^k} (mod n^{r+1}) for r=1, 2, any positive integer m with n \equiv \pm 1 (mod m) in terms of Bernoulli polynomials. As an application, we also obtain some new congruences involving binomial coefficients modulo n^4 in terms of generalized Bernoulli numbers.
