On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order

Abstract

We give some interesting relationships between the power sums and the generalized Bernoulli numbers attached to of higher order using multivariate -adic invariant integral on .

1. Introduction

Let be a fixed prime number. Throughout this paper, the symbols , , , and denote the ring of rational integers, the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. Let be the set of natural numbers, and . Let be the normalized exponential valuation of with (see [1–24]). Let be the space of uniformly differentiable function on . Let be a fixed positive integer. For , let

(1.1)

where lies in . For , the -adic invariant integral on is defined as

(1.2)

(see [11–19]). From (1.2), we note that

(1.3)

where and . Let . Then we can derive the following equation from (1.3):

(1.4)

(see [1–11]). Let be the Dirichlet's character with conductor . Then the generalized Bernoulli polynomials attached to are defined as

(1.5)

and the generalized Bernoulli numbers attached to , , are defined as (see [1–20, 25]). The purpose of this paper is to derive some identities of symmetry for the generalized Bernoulli polynomials attached to of higher order.

2. Symmetric Properties for the Generalized Bernoulli Polynomials of Higher Order

Let be the Dirichlet's character with conductor . Then we note that

(2.1)

where are the th generalized Bernoulli numbers attached to (see [7, 9, 15, 25]). Now we also see that the generalized Bernoulli polynomials attached to are given by

(2.2)

By (2.1) and (2.2), we have

(2.3)

(see [15, 25]), and

(2.4)

(see [1–19, 25]). For , we obtain that

(2.5)

where . Thus, we have

(2.6)

Then

(2.7)

Let us define the -adic function as follows:

(2.8)

(see [25]). By (2.7) and (2.8), we see that

(2.9)

(see [25]). Thus, we have

(2.10)

This means that

(2.11)

(see [25]).

The generalized Bernoulli polynomials attached to of order , which is denoted by , are defined as

(2.12)

Then the values of at are called the generalized Bernoulli numbers attached to of order . When , the polynomials of numbers are called the generalized Bernoulli polynomials or numbers attached to . Let . Then we set

(2.13)

where

(2.14)

In (2.13), we note that is symmetric in . From (2.13), we derive

(2.15)

It is easy to see that

(2.16)

From (2.16), we note that

(2.17)

By the symmetry of in and , we see that

(2.18)

By comparing the coefficients on the both sides of (2.17) and (2.18), we see the following theorem.

Theorem 2.1.

For , one has

(2.19)

Remark 2.2.

Let and in (1.4). Then we have

(2.20)

(see [25]).

We also calculate that

(2.21)

From the symmetric property of in and , we derive

(2.22)

By comparing the coefficients on the both sides of (2.21) and (2.22), we obtain the following theorem.

Theorem 2.3.

For , one has

(2.23)

Remark 2.4.

Let and in (2.23). We have

(2.24)

(see [25]).