`Abstract and Applied AnalysisVolume 2012, Article ID 847901, 12 pageshttp://dx.doi.org/10.1155/2012/847901`
Review Article

## Some Properties and Identities of Bernoulli and Euler Polynomials Associated with p-adic Integral on

1Department of Mathematics, Sogang University, Seoul, Republic of Korea
2Department of Mathematics, Kwangwoon University, Seoul, Republic of Korea
3Hanrimwon, Kwangwoon University, Seoul, Republic of Korea
4Division of General Education, Kwangwoon University, Seoul, Republic of Korea
5Department of Mathematics Education, Kyungpook National University, Taegu, Republic of Korea

Received 15 December 2011; Accepted 8 February 2012

Copyright © 2012 D. S. 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 investigate some properties and identities of Bernoulli and Euler polynomials. Further, we give some formulae on Bernoulli and Euler polynomials by using p-adic integral on .

#### 1. Introduction

Let be a fixed odd prime number. Throughout this paper, , , and will denote the ring of -adic rational integers, the field of -adic rational numbers, and the completion of the algebraic closure of . Let be the normalized exponential valuation of with .

For , the -adic invariant integral on in the bosonic sense is defined by (see [1, 2]). The fermionic -adic integral on is defined by Kim as follows: (see [3]). As is well known, Bernoulli polynomials are defined by with the usual convention about replacing by , symbolically (see [119]). In the special case , is called the th Bernoulli number.

The Euler polynomials are also defined by the generating function as follows: with the usual convention about replacing by , symbolically (see [119]). In the special case , is called the -th Euler number.

By (1.3) and (1.4), we easily see that where (see [14, 16, 19]).

The following properties of Bernoulli numbers and polynomials are well known (see [10, 11]).

For , where is Gauss’ symbol.

First, we investigate some identities of Euler polynomials corresponding to (1.6), (1.7) and (1.8). From those identities, we derive some interesting identities and properties by using -adic integral on .

#### 2. Some Identities of Bernoulli and Euler Polynomials

By (1.4), we get From (2.1), we note that Thus, we have Replacing by in (2.3), we obtain the following proposition.

Proposition 2.1. For , one has

Let us replace by in Proposition 2.1. Then we have Thus, we see that Therefore, adding (2.4) and (2.6), we obtain the following proposition.

Proposition 2.2. For , one has

From (2.2), we note that By (2.3) and (2.8), we get Therefore, replacing by , we obtain the following proposition.

Proposition 2.3. For , one has

Letting in Proposition 2.1, we have Therefore, by (2.11) and (2.12), we obtain the following corollary.

Corollary 2.4. For , one has

Replacing by 1 and by in Proposition 2.2, we have Therefore, by (2.14), we obtain the following corollary.

Corollary 2.5. For , one has

Replacing by 1 and by in Proposition 2.3, we have Therefore, by (2.16), we obtain the following corollary.

Corollary 2.6. For , one has

Replacing by and by in Proposition 2.3, we get Thus, we have Therefore, by (2.19) and (2.20), we obtain the following corollary.

Corollary 2.7. For , we have

Replacing by 1 and by in Proposition 2.2, we get Therefore, by (2.22), we obtain the following corollary.

Corollary 2.8. For , one has

Replacing by and by 1 in Proposition 2.3, we get Therefore, by (2.24), we obtain the following corollary.

Corollary 2.9. For , we have

Replacing by and by in Proposition 2.2, we have Thus, by multipling on both sides, we get By (2.20) and (2.27), we see that Therefore, by (2.28), we obtain the following corollary.

Corollary 2.10. For , we have

From (1.6), we can derive the following equation: Let us take the -adic integral on both sides in (2.30) as follows: for , On the other hand, Therefore, by (2.31) and (2.32), we obtain the following theorem.

Theorem 2.11. For , one has

In (2.30), let us take the fermionic -adic integral on both sides as follows: On the other hand Therefore, by (2.34) and (2.35), we obtain the following theorem.

Theorem 2.12. For , one has

From (1.7), we can easily derive the following equation: Let us take on both sides in (2.37). Then we have On the other hand, where is a Kronecker symbol.

Therefore, by (2.38) and (2.39), we obtain the following theorem.

Theorem 2.13. For , one has

Taking on both sides in (2.37), we get On the other hand Therefore, by (2.41) and (2.42), we obtain the following theorem.

Theorem 2.14. For , one has

From (1.8), we can also derive the following equation: Let us take the bosonic -adic integral on both sides in (2.44). Then we get On the other hand, Therefore, by (2.45) and (2.46), we obtain the following theorem.

Theorem 2.15. For , one has

Now, let us consider the fermionic -adic integral on both sides in (2.44): On the other hand, Therefore, by (2.48) and (2.49), we obtain the following theorem.

Theorem 2.16. For , one has

#### Acknowledgment

The first author was supported by National Research Foundation of Korea Grant funded by the Korean Government 2011-0002486.

#### References

1. T. Kim, “q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,” Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008.
2. T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.
3. 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.
4. S. Araci, D. Erdal, and D.-J. Kang, “Some new properties on the q-Genocchi numbers and polynomials associated with q-Bernstein polynomials,” Honam Mathematical Journal, vol. 33, no. 2, pp. 261–270, 2011.
5. A. Bayad, T. Kim, B. Lee, and S.-H. Rim, “Some identities on Bernstein polynomials associated with q-Euler polynomials,” Abstract and Applied Analysis, vol. 2011, Article ID 294715, 10 pages, 2011.
6. A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.
7. L. Carlitz, “Some arithmetic properties of generalized Bernoulli numbers,” Bulletin of the American Mathematical Society, vol. 65, pp. 68–69, 1959.
8. L. Carlitz, “Note on the integral of the product of several Bernoulli polynomials,” Journal of the London Mathematical Society, vol. 34, pp. 361–363, 1959.
9. M. Cenkci, Y. Simsek, and V. Kurt, “Multiple two-variable p-adic q-L-function and its behavior at $s=0$,” Russian Journal of Mathematical Physics, vol. 15, no. 4, pp. 447–459, 2008.
10. H. Cohen, Number Theory. Vol. II. Analytic and Modern Tools, vol. 240 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2007.
11. H. Cohen, Number Theory. Vol. I. Tools and Diophantine Equations, vol. 239 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2007.
12. D. Ding and J. Yang, “Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 1, pp. 7–21, 2010.
13. 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.
14. 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.
15. H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation functions of higher-order q-Euler numbers and their applications,” Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008.
16. C. S. Ryoo, “Some identities of the twisted q-Euler numbers and polynomials associated with q-Bernstein polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 2, pp. 239–248, 2011.
17. C. S. Ryoo, “Some relations between twisted q-Euler numbers and Bernstein polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 217–223, 2011.
18. 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.
19. Y. Simsek, “Theorems on twisted L-function and twisted Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 11, no. 2, pp. 205–218, 2005.