Research Article | Open Access

H.-M. Kim, D. S. Kim, "Arithmetic Identities Involving Bernoulli and Euler Numbers", *International Journal of Mathematics and Mathematical Sciences*, vol. 2012, Article ID 689797, 10 pages, 2012. https://doi.org/10.1155/2012/689797

# Arithmetic Identities Involving Bernoulli and Euler Numbers

**Academic Editor:**A. Bayad

#### Abstract

The purpose of this paper is to give some arithmatic identities for the Bernoulli and Euler numbers. These identities are derived from the several -adic integral equations 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 algebraic closure of , respectively. The -adic norm is normalized so that . Let be the set of natural numbers and .

Let be the space of uniformly differentiable functions on . For , the bosonic -adic integral on is defined by and the fermionic -adic integral on is defined by Kim as follows (see [1–8]):

The Euler polynomials, , are defined by the generating function as follows (see [1–16]): In the special case, , is called the th Euler number.

By (1.3) and the definition of Euler numbers, we easily see that with the usual convention about replacing by (see [10]). Thus, by (1.3) and (1.4), we have where is the Kronecker symbol (see [9, 10, 17–19]).

From (1.2), we can also derive the following integral equation for the fermionic -adic integral on as follows: see [1, 2]. By (1.3) and (1.6), we get Thus, by (1.7), we have see [1–8, 13–16].

The Bernoulli polynomials, , are defined by the generating function as follows: see [18]. In the special case, , is called the th Bernoulli number. From (1.9) and the definition of Bernoulli numbers, we note that see [1–19], with the usual convention about replacing by . By (1.9) and (1.10), we easily see that see [13].

From (1.1), we can derive the following integral equation on : where and .

By (1.12), we have Thus, by (1.13), we can derive the following Witt’s formula for the Bernoulli polynomials:

In [19], it is known that for , where if or .

The purpose of this paper is to give some arithmetic identities involving Bernoulli and Euler numbers. To derive our identities, we use the properties of -adic integral equations on .

#### 2. Arithmetic Identities for Bernoulli and Euler Numbers

Let us take the bosonic -adic integral on in (1.15) as follows: On the other hand, we get By (2.1) and (2.2), we get

Therefore, by (2.3), we obtain the following theorem.

Theorem 2.1. *For , one has
*

Now we consider the fermionic -adic integral on in (1.15) as follows: On the other hand, we get By (2.5) and (2.6), we get Therefore, by (2.7), we obtain the following theorem.

Theorem 2.2. *For , one has
*

Replacing by in (1.15), we have the identity: Let us take the bosonic -adic integral on in (2.9) as follows:

On the other hand, we see that By (2.10) and (2.11), we get Therefore, by (2.12), we obtain the following theorem.

Theorem 2.3. *For , one has
*

We consider the fermionic -adic integral on in (2.9) as follows: On the other hand, we get By (2.14) and (2.15), we obtain the following theorem.

Theorem 2.4. *For , one has
*

#### Acknowledgment

This Research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2012R1A1A2003786).

#### References

- 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 Site | Google Scholar - T. Kim, “Symmetry of power sum polynomials and multivariate fermionic
*p*-adic invariant integral on ${\mathbb{Z}}_{p}$,”*Russian Journal of Mathematical Physics*, vol. 16, no. 1, pp. 93–96, 2009. View at: Publisher Site | Google Scholar - 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. View at: Google Scholar | Zentralblatt MATH - T. Kim, “
*q*-Volkenborn integration,”*Russian Journal of Mathematical Physics*, vol. 9, no. 3, pp. 288–299, 2002. View at: Google Scholar | Zentralblatt MATH - T. Kim, B. Lee, S. H. Lee, and S.-H. Rim, “Identities for the Bernoulli and Euler numbers and polynomials,”
*Ars Combinatoria*. In press. View at: Google Scholar - S.-H. Rim and J. Jeong, “On the modified
*q*-Euler numbers of higher order with weight,”*Advanced Studies in Contemporary Mathematics*, vol. 22, no. 1, pp. 93–98, 2012. View at: Google Scholar - S.-H. Rim and T. Kim, “Explicit
*p*-adic expansion for alternating sums of powers,”*Advanced Studies in Contemporary Mathematics*, vol. 14, no. 2, pp. 241–250, 2007. View at: Google Scholar - 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. View at: Google Scholar - L. Carlitz, “Some arithmetic properties of generalized Bernoulli numbers,”
*Bulletin of the American Mathematical Society*, vol. 65, pp. 68–69, 1959. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. Carlitz, “Note on the integral of the product of several Bernoulli polynomials,”
*Journal of the London Mathematical Society Second Series*, vol. 34, pp. 361–363, 1959. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Choi, D. S. Kim, T. Kim, and Y. H. Kim, “Some arithmetic identities on Bernoulli and Euler numbers arising from the
*p*-adic integrals on ${\mathbb{Z}}_{p}$,”*Advanced Studies in Contemporary Mathematics*, vol. 22, no. 2, pp. 239–247, 2012. View at: Google Scholar - D. V. Dolgy, T. Kim, B. Lee, and C. S. Ryoo, “On the
*q*-analogue of Euler measure with weight $\alpha $,”*Advanced Studies in Contemporary Mathematics*, vol. 21, no. 4, pp. 429–435, 2011. View at: Google Scholar - D. S. Kim, T. Kim, D. V. Dolgy, S. H. Lee, and S.-H. Rim, “Some properties and identities of Bernoulli and Euler polynomials associated with
*p*-adic integral on ${\mathbb{Z}}_{p}$,”*Abstract and Applied Analysis*, vol. 2012, Article ID 847901, 12 pages, 2012. View at: Publisher Site | Google Scholar - D. S. Kim, N. Lee, J. Na, and K. H. Park, “Identities of symmetry for higher-order Euler polynomials in three variables (I),”
*Advanced Studies in Contemporary Mathematics*, vol. 22, no. 1, pp. 51–74, 2012. View at: Google Scholar - H.-M. Kim, D. S. Kim, T. Kim, S. H. Lee, D. V. Dolgy, and B. Lee, “Identities for the Bernoulli and Euler numbers arising from the
*p*-adic integral on ${\mathbb{Z}}_{p}$,”*Proceedings of the Jangjeon Mathematical Society*, vol. 15, no. 2, pp. 155–161, 2012. View at: Google Scholar - 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. View at: Google Scholar | Zentralblatt MATH - S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionic
*p*-adic*q*-integral on ${\mathbb{Z}}_{p}$ associated with weighted q-Bernstein and*q*-Genocchi polynomials,,”*Abstract and Applied Analysis*, vol. 2011, Article ID 649248, 10 pages, 2011. View at: Publisher Site | Google Scholar - 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 Site | Google Scholar - H. Cohen,
*Number Theory*, vol. 239 of*Graduate Texts in Mathematics*, Springer, New York, NY, USA, 2007.

#### Copyright

Copyright © 2012 H.-M. Kim and D. S. Kim. 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.