Let be the -dimensional vector space over . We show that is a good basis for the space , for our purpose of arithmetical and combinatorial applications. Thus, if is of degree , then for some uniquely determined . In this paper we develop method for computing from the information of .
1. Introduction
The Euler polynomials, , are given by
(see [1–20]) with the usual convention about replacing by . In the special case, are called the th Euler numbers. The Bernoulli numbers are also defined by
(see [1–20]) with the usual convention about replacing by . As is well known, the Bernoulli polynomials are given by
(see [9–15]) From (1.1), (1.2), and (1.3), we note that
where is the kronecker symbol.
Let with . The formula
is proved in [4–6]. Let be the -dimensional vector space over . Probably, is the most natural basis for this space. But is also a good basis for the space , for our purpose of arithmetical and combinatorial applications. Thus, if is of degree , then
for some uniquely determined . Further,
where . In this paper we develop methods for computing from the information of . Apply these results to arithmetically and combinatorially interesting identities involving . Finally, we give some applications of those obtained identities.
2. Euler Basis, Identities, and Their Applications
Let us take the polynomial of degree as follows:
From (2.1), we have
By (1.7) and (2.2), we get
Thus, we have
By (1.6), (2.1), (2.3), and (2.4), we get
Let us consider the following triple identities:
where the sum runs over all with . Thus, by (2.7), we get
From (1.7) and (2.8), we have
Therefore, by (2.7) and (2.9), we obtain the following theorem.
Theorem 2.1. For , and with , one has
Let us take the polynomial as follows:
Then, by (2.11), we get
From (1.6), (1.7), and (2.12), we have
Note that
Therefore, we obtain the following theorem.
Theorem 2.2. For with , one has
Remark 2.3. By the same method, we obtain the following identities. (I)
(II)
Let us consider the polynomial as follows:
Thus, by (2.18), we get
From (1.6), (1.7), (2.18), and (2.19), we have
Here we note that
It is easy to show that
Therefore, by (1.6), (2.18), (2.20), and (2.22), we obtain the following theorem.
Theorem 2.4. For with , one has
Remark 2.5. By the same method, we can derive the following identities. (I)
(II)
Now we generalize the above consideration to the completely arbitrary case. Let
where the sum runs over all nonnegative integers satisfying . From (2.26), we note that
By (1.6), (1.7), (2.18), and (2.27), we get
Note that
Therefore, by (1.6), (2.28), and (2.29), we obtain the following theorem.
Theorem 2.6. For with , one has
Let us consider the polynomial of degree as
Then, from (2.31), we have
By (1.7) and (2.32), we get
From (2.33), we can derive the following equation:
Observe now that
Therefore, by (1.6), (2.31), (2.34), (2.35), and (2.36), we obtain the following theorem.
Theorem 2.7. For with , one has
Let us consider the following polynomial of degree .
Thus, by (2.38), we get
From (1.7), we have
Thus, by (2.40), we get
Now, we note that
Therefore, by (1.6), (2.38), (2.41), and (2.42), we obtain the following theorem.
Theorem 2.8. For with , one has
By the same method, we can obtain the following identity:
Acknowledgments
This paper 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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