Abstract and Applied Analysis

Volume 2012 (2012), Article ID 674210, 10 pages

http://dx.doi.org/10.1155/2012/674210

## Identities Involving -Bernoulli and -Euler Numbers

^{1}Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea^{2}Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea^{3}Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 18 January 2012; Revised 24 February 2012; Accepted 24 February 2012

Academic Editor: Toka Diagana

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 give some identities on the *q*-Bernoulli and *q*-Euler numbers by using *p*-adic integral equations on .

#### 1. Introduction

Let be a fixed odd prime number. Throughout this paper, , , and will denote 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 . The -adic norm is normally defined by .

As it is well known, the Euler polynomials are defined by with the usual convention about replacing by (see [1–14]). In the special case, , is called the th Euler number.

The ordinary Bernoulli polynomials are also defined by with the usual convention about replacing by h (see [1–14]). In the special case, , is called the th Bernoulli number.

Let be the space of uniformly differentiable functions on . For , the bosonic -adic integral on is defined by (see [1, 7]). Let be the translation of with . From (1.3) we have (see [1, 7]).

The fermionic -adic integral on is also defined by T. Kim as follows: (see [6, 15, 16]). By (1.5), we get (see [6, 8]).

Let with and . From (1.4), we have Thus, by (1.7), we see that By (1.8), we get As an indeterminate, let us assume that with .

From (1.1) and (1.6), we note that the -Euler polynomials are given by where are called the th -Euler polynomials (see [1, 3, 6, 8]).

Thus, by (1.10), we get In the special case, , is called the th -Euler number (see [8, 9]). By (1.10) and (1.11), we get the recurrence formula for the -Euler numbers as follows: with the usual convention about replacing by . Here is the -number of and is the Kronecker symbol (see [10, 11]).

From (1.2), (1.7), and (1.8), we have with the usual convention about replacing by (see [1, 3, 14]).

From (1.11), we easily see that (see [14]).

In this paper we give some interesting properties of -adic integrals on associated with the -Bernoulli and the -Euler numbers. From those properties, we derive new identities involving the -Bernoulli and the -Euler numbers arising from -adic integrals of polynomial identities.

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

Let be the cyclic group of order with . Then is defined by the direct limit as . In this section, we assume that , then . From (1.4), we can derive the following equation (2.1): where is called the th -Bernoulli polynomial (see [7]). In the special case, , is called the th -Bernoulli number.

By (2.1), we get with the usual convention about replacing by (see [7, 14]).

From (1.3), we have By (2.3), we get From (2.1) and (2.4), we have By using (1.4), we see that Thus, by (2.1) and (2.6), we get Therefore, by (2.5) and (2.7), we obtain the following theorem.

Theorem 2.1. *For , one has
*

From (1.5) and (1.6), we note that Therefore, by (1.11) and (2.9), we obtain the following theorem.

Theorem 2.2. *For , one has
*

By (1.6), we get Thus, by (1.11) and (2.11), we have Therefore, by Theorem 2.2 and (2.12), we obtain the following theorem.

Theorem 2.3. *For , one has
*

By using the -adic integrals on , we have the following equation (2.14): By (2.14), we get It is not difficult to show that Therefore, by (2.15) and (2.16), we obtain the following theorem.

Theorem 2.4. *For , one has
*

By (2.5), (2.7), (2.12), Theorems 2.1, and 2.3, we get From (2.1), we have Let From (2.18), (2.20), and (2.21), we note that By (2.5), we get By (2.3), we easily see that where is the Kronecker symbol.

Thus, by (2.23) and (2.24), we get By (2.22) and (2.25), we get Therefore, by (2.21) and (2.26), we obtain the following theorem.

Theorem 2.5. *For , one has
*

Let us consider the following integral: By (2.19), we get From Theorem 2.2, we note that By (1.12), we get Thus, by (2.30) and (2.31), we get From (2.29) and (2.32), we note that Therefore, by (2.28) and (2.33), we obtain the following theorem.

Theorem 2.6. *For , one has
*

Now we consider the fermionic -adic integral on for the th -Euler polynomials as follows: On the other hand, by Theorem 2.2, we get From (2.32) and (2.36), we note that Therefore, by (2.35) and (2.37), we obtain the following theorem.

Theorem 2.7. *For , one has
*

From (2.1) and (2.7), we note that Let us consider the following fermionic -adic integral on : Therefore, by (2.40), we obtain the following theorem.

Theorem 2.8. *For , one has
*

From (1.10) and (2.12), we note that Thus, by (2.42), we get Thus, by (2.43), we have

#### Acknowledgments

The authors would like to express their sincere gratitude to referees for their valuable suggestions and comments. The first author was supported by National Research Foundation of Korea Grant funded by the Korean Government 2011-0002486.

#### References

- 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. View at Publisher · View at Google Scholar - A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,”
*Advanced Studies in Contemporary Mathematics*, vol. 20, no. 3, pp. 389–401, 2010. View at Google Scholar - 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. View at Publisher · View at Google Scholar - L. Carlitz, “The product of two Eulerian polynomials,”
*Mathematics Magazine*, vol. 36, no. 1, pp. 37–41, 1963. View at Google Scholar - L. C. Jang, “A note on Nörlund-type twisted
*q*-Euler polynomials and numbers of higher order associated with fermionic invariant*q*-integrals,”*Journal of Inequalities and Applications*, vol. 2010, Article ID 417452, 12 pages, 2010. View at Publisher · View at Google Scholar - T. Kim, “Some identities on the
*q*-Euler polynomials of higher order and*q*-Stirling numbers by the fermionic*p*-adic integral on ${\mathbb{Z}}_{p}$,”*Russian Journal of Mathematical Physics*, vol. 16, no. 4, pp. 484–491, 2009. View at Publisher · View at Google Scholar - T. Kim, “An analogue of Bernoulli numbers and their congruences,”
*Reports of the Faculty of Science and Engineering. Saga University. Mathematics*, vol. 22, no. 2, pp. 21–26, 1994. View at Google Scholar - T. Kim, “New approach to
*q*-Euler polynomials of higher order,”*Russian Journal of Mathematical Physics*, vol. 17, no. 2, pp. 218–225, 2010. View at Publisher · View at Google Scholar - Y.-H. Kim, K.-W. Hwang, and T. Kim, “Interpolation functions of the
*q*-Genocchi and the*q*-Euler polynomials of higher order,”*Journal of Computational Analysis and Applications*, vol. 12, no. 1-B, pp. 228–238, 2010. View at Google Scholar - H. Ozden and Y. Simsek, “A new extension of
*q*-Euler numbers and polynomials related to their interpolation functions,”*Applied Mathematics Letters*, vol. 21, no. 9, pp. 934–939, 2008. View at Publisher · View at Google Scholar - 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. View at Google Scholar - K.-H. Park, Y.-H. Kim, and T. Kim, “A note on the generalized
*q*-Euler numbers (2),”*Journal of Computational Analysis and Applications*, vol. 12, no. 3, pp. 630–636, 2010. View at Google Scholar - 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. View at Google Scholar - Y. Simsek, “On
*p*-adic twisted*p*-*L*-functions related to generalized twisted Bernoulli numbers,”*Russian Journal of Mathematical Physics*, vol. 13, no. 3, pp. 340–348, 2006. View at Publisher · View at Google Scholar - T. Kim, “A note on
*q*-Volkenborn integration,”*Proceedings of the Jangjeon Mathematical Society*, vol. 8, no. 1, pp. 13–17, 2005. View at Google Scholar - T. Kim, “
*q*-Euler numbers and polynomials associated with*p*-adic*q*-integrals,”*Journal of Nonlinear Mathematical Physics*, vol. 14, no. 1, pp. 15–27, 2007. View at Publisher · View at Google Scholar