Bernoulli Basis and the Product of Several Bernoulli Polynomials
Dae San Kim1and Taekyun Kim2
Academic Editor: Yilmaz Simsek
Received25 Jun 2012
Accepted09 Aug 2012
Published08 Sept 2012
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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