`Abstract and Applied AnalysisVolume 2012, Article ID 784307, 15 pageshttp://dx.doi.org/10.1155/2012/784307`
Research Article

## Some Formulae for the Product of Two Bernoulli and Euler Polynomials

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

Received 6 March 2012; Accepted 23 April 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 formulae for the product of two Bernoulli and Euler polynomials arising from the Euler and Bernoulli basis polynomials.

#### 1. Introduction

As is well known, the Bernoulli polynomials are defined by the generating function as follows: (see [121]), with the usual convention about replacing by . In the special case, , are called the th Bernoulli numbers. The Euler polynomials are also defined by the generating function as follows: (see [611]), with the usual convention about replacing by . In the special case, , are called the th Euler numbers. From (1.1) and (1.2), we can derive the following recurrence relations for the Bernoulli and Euler numbers: where is the Kronecker symbol. By (1.1) and (1.2), we get From (1.4), we can derive By (1.4) and (1.5), we get It is easy to show that Thus, we have (see [1118]). By the definition of the Euler polynomials, we get From (1.9), we have (see [118]). By (1.8) and (1.10), we see that the set and are the basis for the space of polynomials of degree less than or equal to with coefficients in (see [121]).

From , let . Then, we note that Let us assume that . Then, we have

Continuing this process, we get Let for . Then, we have Assume that . Then, we get

Continuing this process, we get By (1.16), we get From the properties of the Bernoulli and Euler basis for the space of the polynomials of degree less than or equal to with coefficients in , we derive some identities for the product of two Bernoulli and Euler polynomials.

#### 2. Some Identities for the Bernoulli and Euler Numbers

Let us consider the polynomial , with . Then, we have From the properties of the Bernoulli basis for the space of polynomials of degree less than or equal to with coefficients in , is given by Thus, by (2.2), we get By (2.1) and (2.2), we get Therefore, by (2.3) and (2.4), we obtain the following theorem.

Theorem 2.1. For , one has

From the properties of the Euler basis for the space of polynomials of degree less than or equal to with coefficients in , is given by By (2.1) and (2.6), we get Therefore, by (2.6) and (2.7), we obtain the following theorem.

Theorem 2.2. For , one has

Let us take polynomial with . Then, we have By the basis set for the space of polynomials of degree less than or equal to with coefficients in , we see that is given by From (2.10), we note that Note that where is the beta function.

From (2.11) and (2.12), we have For , by (2.9) and (2.10), we get Therefore, by (2.13) and (2.14), we obtain the following theorem.

Theorem 2.3. For , one has

From the Euler basis for the space of polynomials of degree less than or equal to with coefficients in , we note that can be written as follows: Thus, we have Therefore, by (2.16) and (2.17), we obtain the following theorem.

Theorem 2.4. For , one has

Let us consider the polynomial . Then, we have It is easy to show that For , we have Therefore, by (2.19) and (2.21), we obtain the following theorem.

Theorem 2.5. For , one has

Remark 2.6. If , by the same method, we get

Let us consider the polynomial . Then, we have It is easy to show that

For , we have Therefore, by (2.24) and (2.26), we obtain the following theorem.

Theorem 2.7. For , one has

Let us consider the polynomial . By the same method, we obtain the following identity: Let us take . Then, the th derivative of is given by where Note that where .

By the properties of the Bernoulli basis for the space of polynomials of degree less than or equal to with coefficients in , is given by Thus, by (2.32), we get

From (2.29), (2.31), and (2.32), we note that From (2.30), we have Therefore, by (2.32), (2.34), and (2.35), we obtain the following theorem.

Theorem 2.8. For , one has

We assume that . For , we have Finally,

By the same method, we obtain the following identity: Let us take . Then, for , we have where .

Note that By the properties of the Bernoulli basis for the space of polynomials of degree less than or equal to with coefficients in , can be written as Thus, by (2.42), we get

It is easy to show that For , one has Therefore, by (2.42), (2.44), and (2.45), we obtain the following theorem.

Theorem 2.9. For , one has

We may assume that . Then, we note that

Thus, we have

By the same method, we obtain the following identity:

#### 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, “Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials,” Mathematics of Computation, vol. 80, no. 276, pp. 2219–2221, 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. A. Bayad and T. Kim, “Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 2, pp. 247–253, 2010.
5. L. Carlitz, “The product of two eulerian polynomials,” Mathematics Magazine, vol. 36, no. 1, pp. 37–41, 1963.
6. 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.
7. I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order $w$-$q$-genocchi numbers,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009.
8. K.-W. Hwang, D. V. Dolgy, T. Kim, and S. H. Lee, “On the higher-order $q$-Euler numbers and polynomials with weight $\alpha$,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 354329, 12 pages, 2011.
9. D. S. Kim and T. Kim, “A study on the integral of the product of several Bernoulli polynomials,” communicated.
10. T. Kim, “An identity of symmetry for the generalized Euler polynomials,” Journal of Computational Analysis and Applications, vol. 13, no. 7, pp. 1292–1296, 2011.
11. T. Kim, “Some formulae for the q-Bernstein polynomials and q-deformed binomial distributions,” Journal of Computational Analysis and Applications, vol. 14, no. 5, pp. 917–933, 2012.
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. 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.
14. T. Kim, “$q$-generalized Euler numbers and polynomials,” Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 293–298, 2006.
15. T. Kim, “$q$-volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.
16. 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.
17. 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.
18. 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.
19. 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.
20. 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.
21. Y. Simsek, “Special functions related to Dedekind-type DC-sums and their applications,” Russian Journal of Mathematical Physics, vol. 17, no. 4, pp. 495–508, 2010.