Some identities of symmetry for the generalized Bernoulli numbers and polynomials

In this paper, by the properties of p-adic invariant integral on Zp, we establish various identities concerning the generalized Bernoulli numbers and polynomials. From the symmetric properties of p-adic invariant integral on Zp, we give some interesting relationship between the power sums and the generalized Bernoulli polynomials.


§1. Introduction
Let p be a fixed prime number. Throughout this paper, the symbols Z, Z p , Q p , and C p will denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Q p , respectively. Let N be the set of natural numbers and Z + = N ∪ {0}. Let v p be the normalized exponential valuation of C p with |p| p = p −vp(p) = 1/p. Let UD(Z p ) be the space of uniformly differentiable function on Z p . For f ∈ UD(Z p ), the p-adic invariant integral on Z p is defined as f (x), (see [6]). (1) From the definition (1), we have Let f n (x) = f (x + n), (n ∈ N). Then we can derive the following equation (3) from (2).
It is well known that the ordinary Bernoulli polynomials B n (x) are defined as  ), and the Bernoulli number B n are defined as B n = B n (0). Let d a fixed positive integer. For n ∈ N, we set where a ∈ Z lies in 0 ≤ a < dp N . In [6], it is known that Let us take f (x) = e tx . Then we have Thus, we note that Zp x n dx = B n , n ∈ Z + , (see ).
Let χ be the Dirichlet's character with conductor d ∈ N. Then the generalized Bernoulli polynomials attached to χ are defined as and the generalized Bernoulli numbers attached to χ, B n,χ are defined as B n,χ = B n,χ (0). In this paper, we investigate the interesting identities of symmetry for the generalized Bernoulli numbers and polynomials attached to χ by using the properties of p-adic invariant integral on Z p . Finally, we will give relationship between the power sum polynomials and the generalized Bernoulli numbers attached to χ. §2. Symmetry of power sum and the generalized Bernoulli polynomials Let χ be the Dirichlet character with conductor d ∈ N. From (3), we note that where B n,χ (x) are n-th generalized Bernoulli numbers attached to χ. Now, we also see that the generalized Bernoulli polynomials attached to χ are given by By (5) and (6), we easily see that From (6), we have From (6), we can also derive Therefore, we obtain the following lemma.
We observe that Thus, we have Let us define the p-adic functional T k (χ, n) as follows: By (10) and (11), we see that By using Taylor expansion in (12), we have That is, Let w 1 , w 2 , d ∈ N. Then we consider the following integral equation From (9) and (12), we note that Let us consider the p-adic functional T χ (w 1 , w 2 ) as follows: Then we see that T χ (w 1 , w 2 ) is symmetric in w 1 and w 2 , and By (16) and (17), we have From the symmetric property of T χ (w 1 , w 2 ) in w 1 and w 2 , we note that By comparing the coefficients on the both sides of (18) and (19), we obtain the following theorem.
Let x = 0 in Theorem 2. Then we have By (15) and (17), we also see that From the symmetric property of T χ (w 1 , w 2 ) in w 1 and w 2 , we can also derive the following equation.
By comparing the coefficients on the both sides of (20) and (21), we obtain the following theorem.
Theorem 3. For w 1 , w 2 , d ∈ N, we have Remark. Let x = 0 in Theorem 3. Then we see that If we take w 2 = 1, then we have χ(i)B k,χ (w 1 i).