Some congruences involving generalized Bernoulli numbers and Bernoulli polynomials

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.

modn r+1 for r ∈ {1, 2}, any positive integer m with n ≡ ±1 (modm) 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.

Introduction
Let n, r ≥ 2 be positive integers with (n, r) = 1 and q r (n) denote the Euler quotient, i.e., where φ is the Euler totient function.For the trivial Dirichlet character χ n modulo n, the generalized Bernoulli numbers {B s,χn } are defined by B s,χn t s s! .
Moreover, we have the identity B s,χn = B s p|n (1 − p s−1 ), where B s is the sth Bernoulli number.On the other hand, Bernoulli polynomials {B s (x)} are defined by

It is interesting to investigate congruences of sums
involving the Euler quotient, generalized Bernoulli numbers and Bernoulli polynomials modulo integer powers.For example, in 1938, Lehmer (see [Leh38]) established the famous congruence (1.1) for any odd prime p. Applying this congruence and other similar congruences, Lehmer obtained some results in studying the Fermat's Last Theorem (see [Rib79]).The proof of (1.1) mainly relied on Bernoulli polynomials of fractional arguments.In 2002, Cai (see [Cai02]) used an identity involving generalized Bernoulli numbers proved by Szmidt, Urbanowicz and Zagier (see [SUZ95,(6)]) and obtained a more general congruence for any odd integer n > 1, that is (1.2) In 2008, Sun (see [Sun08a]) determined the following congruence (1.3) for prime p > 5, where E n is the nth Euler number (see [Sun08b]).In 2012, Kanemitsu, Urbanowicz and Wang (see [KUW12]) generalized Sun's congruence (1.3) by using Cai's technique and showed that (1.4) for positive odd integer n > 3. Later in 2015, Kanemitsu, Kuzumaki and Urbanowicz (see [KKU14]) also adopted the identity proved in [SUZ95] and obtained some new congruences of the sums x k (modn r+1 ) for r ∈ {0, 1, 2}, all divisors m of 24.Recently in 2019, Cai, Zhong and Chern (see x k not only play an important role in studying the first case of Fermat's last theorem, but also help to generalize some congruences involving binomial coefficients. For example, in 2002, using (1.2), Cai (see [Cai02, Theorem 2]) established the following congruence for an arbitrary positive integer n > 1, which is a generalization of the well-known Morley's congruence (see [Mor94]) for any prime p ≥ 5. Morley's congruence has a profound impact in combinatorial number theory.In 2008, applying (1.3),Sun (see [Sun08a, Theorem 3.8]) proved for prime p > 5.In 2019, Cai, Zhong and Chern (see [CZC19, Theorem 1.2]) showed that (1.7) for any positive integer k and odd integer n > 1.
In this paper, we study by applying the identity proved by Sun (see [Sun08b, Theorem 2.1]), which is where p, m ∈ N, k, r ∈ Z with k 0 and {x} is the decimal part of x.We get congruences of x k (modn r+1 ) for r ∈ {1, 2}, any positive integer m ≥ 2 with n ≡ ±1 (modm).Moreover, we use these congruences to get some new congruences involving binomial coefficients modulo n 4 .For example, for any positive integers k ≥ 1, n > 1 with (n, 6) = 1, we have and where v is a positive odd integer.At last, using T 1,k (n) (modn r+1 ) for r ∈ {1, 2}, we have where n > 1 is a positive integer with (n, 6) = 1, u, v are positive integers with u > v.

Basic lemmas
In this section, we introduce the following lemmas will be used later.We begin with the divisibility of Bernoulli polynomials.
Proof.we recall an identity given by Sun, Z. W. (see Changing k to l, similarly, we get the following congruence , we can easily get Lemma 2.5 from Lemma 2.1 and Euler's theorem.
Lemma 2.6.Let n > 1 be a positive integer with Proof.By Euler's theorem we have and the von Staudt-Clausen theorem, we see that Noting that B 2n+1 = 0 (n ≥ 1), we can complete the proof of Lemma 2.6.

Theorems
In order to express the following theorems briefly, we denote where J m (n) is the Jacobi symbol for any integers n, m. where Proof.Let n = dp l with prime p ≥ 5, p ∤ d, l ≥ 1.Since n ≡ ±1 (modm), the least positive residue of n modulo m is 1 or m − 1.By Lemma 2.2(ii) and Lemma 2.4, we get Assume q 1 , q 2 , . . ., q g are different prime factors of d.
From Lemma 2.5 and Lemma 2.1, we observe and Applying (3.2), (3.3), (3.4) to (3.1), we can get When 2 ∤ k, (n, k + 1) = 1, by Lemma 2.5 and Lemma 2.1 again, we see and Using the similar method of the first case, we can obtain the following congruence and complete the proof of Theorem 3.1.
Corollary 3.2.Let n > 1 be a positive integer with (n, 6) = 1, we have Proof.By Lemma 2.3, 3 φ(n) = 1 + nq 3 (n) and (3.5), we can get Taking m = 3 and k = 1 in Theorem 3.1, then using the above congruence, Lemma 2.3 and the von Staudt-Clausen theorem, (i) is obtained.Taking m = 3 and k = 2 in Theorem 3.1, by Lemma 2.3 and the the von Staudt-Clausen theorem again, we complete the proof of (ii).
Corollary 3.3.Let n > 1 be a positive integer with (n, 6) = 1, then Proof.By Lemma 2.3, 2 φ(n) = 1 + nq 2 (n) and (3.5), we have Taking m = 4 and k = 1 in Theorem 3.1, then using the above congruence, Taking m = 6, k = 1 and m = 6, k = 2 in Theorem 3.1, and using the above congruence, we get Corollary 3.4(i) and (ii) respectively in a way similar to Corollary 3.3. where Proof.Let n = dp l with prime p ≥ 3, p ∤ d, l ≥ 1, By Lemma 2.4, we have From Lemma 2.5 and Lemma 2.1, we see that when and and Let s = φ(n 2 ) − k.Theorem 3.2 follows from a similar argument of theorem 3.1.
Using the above theorems and corollaries for m = 2, we have the following results.
On the other hand, we also have .