Abstract and Applied Analysis

Volume 2012 (2012), Article ID 784307, 15 pages

doi:10.1155/2012/784307

Research Article

Some Formulae for the Product of Two Bernoulli and Euler Polynomials

D. S. Kim,¹ D. V. Dolgy,² T. Kim,³ and S.-H. Rim⁴

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

Academic Editor: Sung Guen Kim

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 [1–21]), 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 [6–11]), 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 [11–18]). By the definition of the Euler polynomials, we get

From (1.9), we have

(see [1–18]). 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 [1–21]).

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:

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