`International Journal of Mathematics and Mathematical SciencesVolume 2012, Article ID 343981, 15 pageshttp://dx.doi.org/10.1155/2012/343981`
Research Article

## Euler Basis, Identities, and Their Applications

1Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 11 June 2012; Accepted 9 August 2012

Copyright © 2012 D. S. Kim and T. 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.

#### Abstract

Let be the -dimensional vector space over . We show that is a good basis for the space , for our purpose of arithmetical and combinatorial applications. Thus, if is of degree , then for some uniquely determined . In this paper we develop method for computing from the information of .

#### 1. Introduction

The Euler polynomials, , are given by (see ) with the usual convention about replacing by . In the special case, are called the th Euler numbers. The Bernoulli numbers are also defined by (see ) with the usual convention about replacing by . As is well known, the Bernoulli polynomials are given by (see ) From (1.1), (1.2), and (1.3), we note that where is the kronecker symbol.

Let with . The formula is proved in . Let be the -dimensional vector space over . Probably, is the most natural basis for this space. But is also a good basis for the space , for our purpose of arithmetical and combinatorial applications. Thus, if is of degree , then for some uniquely determined . Further, where . In this paper we develop methods for computing from the information of . Apply these results to arithmetically and combinatorially interesting identities involving . Finally, we give some applications of those obtained identities.

#### 2. Euler Basis, Identities, and Their Applications

Let us take the polynomial of degree as follows: From (2.1), we have By (1.7) and (2.2), we get Thus, we have By (1.6), (2.1), (2.3), and (2.4), we get Let us consider the following triple identities: where the sum runs over all with . Thus, by (2.7), we get From (1.7) and (2.8), we have Therefore, by (2.7) and (2.9), we obtain the following theorem.

Theorem 2.1. For , and with , one has

Let us take the polynomial as follows: Then, by (2.11), we get

From (1.6), (1.7), and (2.12), we have Note that Therefore, we obtain the following theorem.

Theorem 2.2. For with , one has

Remark 2.3. By the same method, we obtain the following identities.
(I)
(II) Let us consider the polynomial as follows: Thus, by (2.18), we get From (1.6), (1.7), (2.18), and (2.19), we have Here we note that It is easy to show that Therefore, by (1.6), (2.18), (2.20), and (2.22), we obtain the following theorem.

Theorem 2.4. For with , one has

Remark 2.5. By the same method, we can derive the following identities.
(I)
(II)

Now we generalize the above consideration to the completely arbitrary case. Let where the sum runs over all nonnegative integers satisfying . From (2.26), we note that

By (1.6), (1.7), (2.18), and (2.27), we get Note that Therefore, by (1.6), (2.28), and (2.29), we obtain the following theorem.

Theorem 2.6. For with , one has

Let us consider the polynomial of degree as Then, from (2.31), we have By (1.7) and (2.32), we get From (2.33), we can derive the following equation: Observe now that Therefore, by (1.6), (2.31), (2.34), (2.35), and (2.36), we obtain the following theorem.

Theorem 2.7. For with , one has

Let us consider the following polynomial of degree . Thus, by (2.38), we get From (1.7), we have Thus, by (2.40), we get Now, we note that Therefore, by (1.6), (2.38), (2.41), and (2.42), we obtain the following theorem.

Theorem 2.8. For with , one has

By the same method, we can obtain the following identity:

#### Acknowledgments

This paper 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

1. S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionic $p$-adic $q$-integral representation on ${ℤ}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials,” Abstract and Applied Analysis, vol. 2011, Article ID 649248, 10 pages, 2011.
2. A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.
3. 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.
4. 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.
5. L. Carlitz, “Multiplication formulas for products of Bernoulli and Euler polynomials,” Pacific Journal of Mathematics, vol. 9, pp. 661–666, 1959.
6. L. Carlitz, “Arithmetic properties of generalized Bernoulli numbers,” Journal für die Reine und Angewandte Mathematik, vol. 202, pp. 174–182, 1959.
7. N. S. Jung, H. Y. Lee, and C. S. Ryoo, “Some relations between twisted $\left(h,q\right)$-Euler numbers with weight $\alpha$ and $q$-Bernstein polynomials with weight $\alpha$,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 176296, 11 pages, 2011.
8. D. S. Kim, “Identities of symmetry for $q$-Euler polynomials,” Open Journal of Discrete Mathematics, vol. 1, no. 1, pp. 22–31, 2011.
9. D. S. Kim, “Identities of symmetry for generalized Euler polynomials,” International Journal of Combinatorics, vol. 2011, Article ID 432738, 12 pages, 2011.
10. T. Kim, “On the weighted $q$-Bernoulli numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 207–215, 2011.
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, “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.
13. B. Kurt and Y. Simsek, “Notes on generalization of the Bernoulli type polynomials,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 906–911, 2011.
14. H. Y. Lee, N. S. Jung, and C. S. Ryoo, “A note on the $q$-Euler numbers and polynomials with weak weight $\alpha$,” Journal of Applied Mathematics, vol. 2011, Article ID 497409, 14 pages, 2011.
15. H. Ozden, “$p$-adic distribution of the unification of the Bernoulli, Euler and Genocchi polynomials,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 970–973, 2011.
16. H. Ozden, I. N. Cangul, and Y. Simsek, “On the behavior of two variable twisted p-adic Euler q-l-functions,” Nonlinear Analysis, vol. 71, no. 12, pp. e942–e951, 2009.
17. S.-H. Rim, A. Bayad, E.-J. Moon, J.-H. Jin, and S.-J. Lee, “A new construction on the $q$-Bernoulli polynomials,” Advances in Difference Equations, vol. 2011, article 34, 2011.
18. 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.
19. Y. Simsek, “Complete sum of products of $\left(h,q\right)$-extension of Euler polynomials and numbers,” Journal of Difference Equations and Applications, vol. 16, no. 11, pp. 1331–1348, 2010.
20. 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.