`Discrete Dynamics in Nature and SocietyVolume 2011, Article ID 856132, 11 pageshttp://dx.doi.org/10.1155/2011/856132`
Research Article

## Some New Identities on the Bernoulli and Euler Numbers

1Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
3Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea
4Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea
5Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea

Received 6 October 2011; Revised 31 October 2011; Accepted 31 October 2011

Academic Editor: Lee-Chae Jang

Copyright © 2011 Dae San Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We give some new identities on the Bernoulli and Euler numbers by using the bosonic p-adic integral on and reflection symmetric properties of Bernoulli and Euler polynomials.

#### 1. Introduction

Let be a fixed prime number. Throughout this paper , and will denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of algebraic closure of . Let be the space of uniformly differentiable functions on . For , the bosonic -adic integral on is defined by From (1.1), we note that see [1]. As is well known, the ordinary Bernoulli polynomials are defined by the generating function as follows: see [119], where we use the technical notation by replacing by , symbolically. In the special case, , are called the -th ordinary Bernoulli numbers. That is, the generating function of ordinary Bernoulli numbers is given by see [119]. From (1.4), we can derive the following relation: see [1, 10], where is the Kronecker symbol.

By (1.3) and (1.4), we easily get By (1.2) and (1.3), we easily get see [1, 10]. From (1.7), we can derive Witt’s formula for the -th Bernoulli polynomials as follows: see [11]. By (1.1) and (1.8), we easily see that Thus, by (1.8) and (1.9), we get reflection symmetric relation for the Bernoulli polynomials as follows: The ordinary Euler polynomials are defined by the generating function as follows: with the usual convention about replacing by (see [8, 9]). In the special case, , are called the -th Euler numbers (see [8, 9]).

From (1.11), we note that By comparing the coefficients on both sides of (1.11) and (1.12), we obtain the following reflection symmetric relation for Euler polynomials as follows: The equations (1.10) and (1.13) are useful in deriving our main results in this paper.

For , the Bernstein polynomials are defined by see [13]. By (1.14), we easily get .

In this paper we consider the -adic integrals for the Bernoulli and Euler polynomials. From those -adic integrals, we derive some new identities on the Bernoulli and Euler numbers.

#### 2. Identities on the Bernoulli and Euler Numbers

First, we consider the -adic integral on for the th ordinary Bernoulli polynomials as follows: On the other hand, by (1.3) and (1.10), one gets From (1.5), (1.6), (1.8), and (2.2), one notes that Equating (2.1) and (2.3), one gets Let with . Then, by (2.4), one has Therefore, by (2.4) and (2.5), we obtain the following theorem.

Theorem 2.1. For, one has In particular,

By the same motivation, let us also consider the -adic integral on for Euler polynomials as follows: On the other hand, by (1.12) and (1.13), one gets From (1.12) and the definition of Euler numbers, one has see [8, 9] with the usual convention of replacing by . By (2.9), (2.10), and (2.11), one gets Equating (2.8) and (2.12), one has Therefore, by (2.13), we obtain the following theorem.

Theorem 2.2. For , one has In particular,

Let us consider the following -adic integral on for the product of Bernoulli and Euler polynomials as follows: On the other hand, by (1.10) and (1.13), one gets Equating (2.16) and (2.17), one gets For , by (2.18), one gets Therefore, by (2.19), one obtains the following theorem.

Theorem 2.3. For , one has In particular, for , one has

By the same motivation, we consider the -adic integral on for the product of Bernoulli and Bernstein polynomials as follows: From (1.6) and (1.14), one gets On the other hand, Equating (2.23) and (2.24), one gets By (2.25), we obtain the following theorem.

Theorem 2.4. For , one has

Now, we consider the -adic integral on for the product of Euler and Bernstein polynomials as follows: On the other hand, by (1.13) and (1.14), one gets Equating (2.27) and (2.28), one gets Therefore, by (2.11) and (2.29), we obtain the following theorem.

Theorem 2.5. For , one has

Finally, we consider the -adic integral on for the product of Euler, Bernoulli, and Bernstein polynomials as follows:

On the other hand, by (1.10), (1.13), and (1.14), one gets Equating (2.31) and (2.32), we easily see that Therefore, by (1.5) and (2.11), we obtain the following theorem.

Theorem 2.6. For , one has

#### Acknowledgments

The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2011.

#### References

1. T. Kim, “Symmetry p-adic invariant integral on ${ℤ}_{p}$ for Bernoulli and Euler polynomials,” Journal of Difference Equations and Applications, vol. 14, no. 12, pp. 1267–1277, 2008.
2. A. Bayad and T. Kim, “Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 2, pp. 133–143, 2011.
3. A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.
4. L. Jang, “A note on Kummer congruence for the Bernoulli numbers of higher order,” Proceedings of the Jangjeon Mathematical Society, vol. 5, no. 2, pp. 141–146, 2002.
5. L. C. Jang and H. K. Pak, “Non-Archimedean integration associated with q-Bernoulli numbers,” Proceedings of the Jangjeon Mathematical Society, vol. 5, no. 2, pp. 125–129, 2002.
6. G. Kim, B. Kim, and J. Choi, “The DC algorithm for computing sums of powers of consecutive integers and Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 137–145, 2008.
7. T. Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on ${ℤ}_{p}$,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009.
8. T. Kim, “Euler numbers and polynomials associated with zeta functions,” Abstract and Applied Analysis, vol. 2008, Article ID 581582, 11 pages, 2008.
9. T. Kim, “Note on the Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 131–136, 2008.
10. T. Kim, “On explicit formulas of p-adic q-L-functions,” Kyushu Journal of Mathematics, vol. 48, no. 1, pp. 73–86, 1994.
11. T. Kim, “Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ${ℤ}_{p}$,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009.
12. T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.
13. T. Kim, “A note on q-Bernstein polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 2, pp. 14–50, 2011.
14. A. Kudo, “A congruence of generalized Bernoulli number for the character of the first kind,” Advanced Studies in Contemporary Mathematics, vol. 2, pp. 1–8, 2000.
15. H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on q-Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.
16. S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin, “On the q-Genocchi numbers and polynomials associated with q-zeta function,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 3, pp. 261–267, 2009.
17. C. S. Ryoo, “On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity,” Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 2, pp. 255–263, 2010.
18. I. Buyukyazici, “On Generalized q-Bernstein Polynomials,” Global Journal of Pure and Applied Mathematics, vol. 6, no. 3, pp. 1331–1348, 2010.
19. Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–278, 2008.