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On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order
Journal of Inequalities and Applications volume 2009, Article number: 640152 (2009)
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
where lies in . For , the -adic invariant integral on is defined as
(see [11–19]). From (1.2), we note that
where and . Let . Then we can derive the following equation from (1.3):
(see [1–11]). Let be the Dirichlet's character with conductor . Then the generalized Bernoulli polynomials attached to are defined as
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
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
By (2.1) and (2.2), we have
(see [1–19, 25]). For , we obtain that
where . Thus, we have
Then
Let us define the -adic function as follows:
(see [25]). By (2.7) and (2.8), we see that
(see [25]). Thus, we have
This means that
(see [25]).
The generalized Bernoulli polynomials attached to of order , which is denoted by , are defined as
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
where
In (2.13), we note that is symmetric in . From (2.13), we derive
It is easy to see that
From (2.16), we note that
By the symmetry of in and , we see that
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
Remark 2.2.
Let and in (1.4). Then we have
(see [25]).
We also calculate that
From the symmetric property of in and , we derive
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
Remark 2.4.
Let and in (2.23). We have
(see [25]).
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Acknowledgment
The present research has been conducted by the research Grant of the Kwangwoon University in 2009.
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Kim, T., Jang, LC., Kim, YH. et al. On the Identities of Symmetry for the Generalized Bernoulli Polynomials Attached to of Higher Order. J Inequal Appl 2009, 640152 (2009). https://doi.org/10.1155/2009/640152
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DOI: https://doi.org/10.1155/2009/640152