International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 463659, 12 pages

doi:10.1155/2012/463659

Research Article

Bernoulli Basis and the Product of Several Bernoulli Polynomials

Dae San Kim¹ and Taekyun Kim²

¹Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

²Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 25 June 2012; Accepted 9 August 2012

Academic Editor: Yilmaz Simsek

Copyright © 2012 Dae San Kim and Taekyun 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.

We develop methods for computing the product of several Bernoulli and Euler polynomials by using Bernoulli basis for the vector space of polynomials of degree less than or equal to .

1. Introduction

It is well known that, the th Bernoulli and Euler numbers are defined by

where and is the Kronecker symbol (see [1–20]).

The Bernoulli and Euler polynomials are also defined by

Note that forms a basis for the space .

So, for a given , we can write

(see [8–12]) for uniquely determined .

Further,

Probably, is the most natural basis for the space . But is also a good basis for the space , for our purpose of arithmetical and combinatorial applications.

What are common to , , ? A few proportion common to them are as follows:

(i)they are all monic polynomials of degree with rational coefficients; (ii), , ; (iii), , .

In [5, 6], Carlitz introduced the identities of the product of several Bernoulli polynomials:

In this paper, we will use (1.4) to derive the identities of the product of several Bernoulli and Euler polynomials.

2. The Product of Several Bernoulli and Euler Polynomials

Let us consider the following polynomials of degree :

where the sum runs over all nonnegative integers ,   satisfying ,  , .

Thus, from (2.1), we have

For , by (1.4), we get

From (2.3), we note that

Therefore, by (1.3), (2.1), (2.3), and (2.4), we obtain the following theorem.

Theorem 2.1. For with , we have

Let us take the polynomial of degree as follows:

Then, from (2.6), we have

By (1.4) and (2.7), we get, for ,

Now, we look at and .

From (2.6), we note that

Therefore, by (1.3), (2.6), (2.8), (2.9), and (2.10), we obtain the following theorem.

Theorem 2.2. For with , one has

Consider the following polynomial of degree :

Then, from (2.12), one has

By (1.4) and (2.13), one gets, for ,

Now look at and :

It is easy to show that

Therefore, by (1.3), (2.14), and (2.15), one obtains the following theorem.

Theorem 2.3. For with , one has

Take the polynomial of degree as follows:

Then, from (2.18), one gets

By (1.4) and (2.19), one gets, for ,

Now look at and :

From (2.18), one can derive the following identity:

Therefore, by (1.3), (2.20), (2.21), and (2.22), one obtains the following theorem.

Theorem 2.4. For with , one has

Acknowledgments

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.

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