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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 463659, 12 pages
Bernoulli Basis and the Product of Several Bernoulli Polynomials
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.
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 .
The Bernoulli and Euler polynomials are also defined by Note that forms a basis for the space .
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 , , .
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
Theorem 2.3. For with , one has
Theorem 2.4. For with , one has
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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