Open Access

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

  • Taekyun Kim1Email author,
  • Lee-Chae Jang2,
  • Young-Hee Kim1 and
  • Kyung-Won Hwang3
Journal of Inequalities and Applications20092009:640152

https://doi.org/10.1155/2009/640152

Received: 5 June 2009

Accepted: 5 August 2009

Published: 26 August 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 [124]). 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 [1119]). From (1.2), we note that
(1.3)
where and . Let . Then we can derive the following equation from (1.3):
(1.4)
(see [111]). 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 [120, 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 [119, 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]).

Declarations

Acknowledgment

The present research has been conducted by the research Grant of the Kwangwoon University in 2009.

Authors’ Affiliations

(1)
Division of General Education-Mathematics, Kwangwoon University
(2)
Department of Mathematics and Computer Science, Konkook University
(3)
Department of General Education, Kookmin University

References

  1. Cenkci M: The -adic generalized twisted -Euler- -function and its applications. Advanced Studies in Contemporary Mathematics 2007,15(1):37–47.MathSciNetMATHGoogle Scholar
  2. Cenkci M, Simsek Y, Kurt V: Multiple two-variable -adic - -function and its behavior at . Russian Journal of Mathematical Physics 2008,15(4):447–459. 10.1134/S106192080804002XMathSciNetView ArticleMATHGoogle Scholar
  3. Jang L-C, Kim S-D, Park D-W, Ro Y-S: A note on Euler number and polynomials. Journal of Inequalities and Applications 2006, 2006:-5.Google Scholar
  4. Kim T: -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russian Journal of Mathematical Physics 2008,15(1):51–57.MathSciNetView ArticleMATHGoogle Scholar
  5. Kim T: A note on -Volkenborn integration. Proceedings of the Jangjeon Mathematical Society 2005,8(1):13–17.MathSciNetMATHGoogle Scholar
  6. Kim T: -Euler numbers and polynomials associated with -adic -integrals. Journal of Nonlinear Mathematical Physics 2007,14(1):15–27. 10.2991/jnmp.2007.14.1.3MathSciNetView ArticleMATHGoogle Scholar
  7. Kim T: A note on -adic -integral on associated with -Euler numbers. Advanced Studies in Contemporary Mathematics 2007,15(2):133–137.MathSciNetMATHGoogle Scholar
  8. Kim T: On the -extension of Euler and Genocchi numbers. Journal of Mathematical Analysis and Applications 2007,326(2):1458–1465. 10.1016/j.jmaa.2006.03.037MathSciNetView ArticleMATHGoogle Scholar
  9. Kim T: On -adic - -functions and sums of powers. Journal of Mathematical Analysis and Applications 2007,329(2):1472–1481. 10.1016/j.jmaa.2006.07.071MathSciNetView ArticleMATHGoogle Scholar
  10. Kim T: On the multiple -Genocchi and Euler numbers. Russian Journal of Mathematical Physics 2008,15(4):481–486. 10.1134/S1061920808040055MathSciNetView ArticleMATHGoogle Scholar
  11. Kim T: New approach to -Euler, Genocchi numbers and their interpolation functions. Advanced Studies in Contemporary Mathematics 2009,18(2):105–112.MathSciNetMATHGoogle Scholar
  12. Kim T: On a -analogue of the -adic log gamma functions and related integrals. Journal of Number Theory 1999,76(2):320–329. 10.1006/jnth.1999.2373MathSciNetView ArticleMATHGoogle Scholar
  13. Kim T: Sums of products of -Euler numbers. to appear in Journal of Computational Analysis and Applications to appear in Journal of Computational Analysis and ApplicationsGoogle Scholar
  14. Kim T, Choi JY, Sug JY: Extended -Euler numbers and polynomials associated with fermionic -adic -integral on . Russian Journal of Mathematical Physics 2007,14(2):160–163. 10.1134/S1061920807020045MathSciNetView ArticleMATHGoogle Scholar
  15. Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002,9(3):288–299.MathSciNetMATHGoogle Scholar
  16. Kim Y-H, Kim W, Jang L-C: On the -extension of Apostol-Euler numbers and polynomials. Abstract and Applied Analysis 2008, 2008:-10.Google Scholar
  17. Pečarić J, Vukelić A: General dual Euler-Simpson formulae. Journal of Mathematical Inequalities 2008,2(4):511–526.MathSciNetMATHGoogle Scholar
  18. Simsek Y: Complete sum of products of -extension of the Euler polynomials and numbers. http://arxiv.org/abs/0707.2849
  19. Simsek Y: On -adic twisted - - functions related to generalized twisted Bernoulli numbers. Russian Journal of Mathematical Physics 2006,13(3):340–348. 10.1134/S1061920806030095MathSciNetView ArticleMATHGoogle Scholar
  20. Simsek Y, Kurt V, Kim D: New approach to the complete sum of products of the twisted -Bernoulli numbers and polynomials. Journal of Nonlinear Mathematical Physics 2007,14(1):44–56. 10.2991/jnmp.2007.14.1.5MathSciNetView ArticleMATHGoogle Scholar
  21. Kim T: Symmetry properties of higher order Bernoulli polynomials. (communicated) (communicated)Google Scholar
  22. Zhang Z, Yang H: Some closed formulas for generalized Bernoulli-Euler numbers and polynomials. Proceedings of the Jangjeon Mathematical Society 2008,11(2):191–198.MathSciNetMATHGoogle Scholar
  23. Tekcan A, Özkoç A, Gezer B, Bizim O: Some relations involving the sums of Fibonacci numbers. Proceedings of the Jangjeon Mathematical Society 2008,11(1):1–12.MathSciNetMATHGoogle Scholar
  24. Khrennikov AYu: Generalized probabilities taking values in non-Archimedean fields and in topological groups. Russian Journal of Mathematical Physics 2007,14(2):142–159. 10.1134/S1061920807020033MathSciNetView ArticleMATHGoogle Scholar
  25. Kim T, Rim S-H, Lee B: Some identities of symmetry for the generalized Bernoulli numbers and polynomials. Abstract and Applied Analysis 2009, 2009:-8.Google Scholar

Copyright

© Taekyun Kim et al. 2009

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