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A note on higher-order Bernoulli polynomials
Journal of Inequalities and Applicationsvolume 2013, Article number: 111 (2013)
Let be the -dimensional vector space over Q. From the property of the basis for the space , we derive some interesting identities of higher-order Bernoulli polynomials.
Let and . For a fixed , the n th Bernoulli polynomials are defined by the generating function to be
with the usual convention about replacing by . In the special case, , are called the n th Bernoulli numbers of order r.
From (1), we note that
Thus, by (2) we get the Euler-type sums of products of Bernoulli numbers as follows:
By (2) and (3), we see that is a monic polynomial of degree n with coefficients in Q.
From (2), we note that
Let Ω denote the space of real-valued differential functions on . Now, we define three linear operators I, △, D on Ω as follows:
Then we see that (i) , (ii) , (iii) .
Let be the -dimensional vector space over Q. 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.
Let . Then can be generated by as follows:
In this paper, we develop methods for uniquely determining from the information of . From those methods, we derive some interesting identities of higher-order Bernoulli polynomials.
2 Higher-order Bernoulli polynomials
For , by (1), we get (). Let .
For a fixed , can be generated by as follows:
From (6) and (7), we can derive the following identities:
By (5) and (8), we get
It is easy to show that
By (7) and (9), we get
From (6) and (12), we note that
Thus, by (13) we get
Hence, from (14) we have
Case 1. Let . Then for all .
By (15), we get
Case 2. Assume that .
For , by (15) we get(17)
For , by (15) we see that(18)
Therefore, by (7), (16), (17) and (18), we obtain the following theorem.
For , we have
For , we have
Let us take . Then can be expressed as a linear combination of . For , we have
Therefore, by Theorem 1 and (19), we obtain the following corollary.
Corollary 2 For with , we have
Let us assume that with . Observe that
Thus, by Theorem 1 and (20), we obtain the following corollary.
Corollary 3 For with , we have
Let us take (). Then can be generated by as follows:
For , we have
Thus, by Theorem 1 and (22), we obtain the following theorem.
Theorem 4 For with , we have
In particular, for , we have
By comparing coefficients on the both sides of (23), we get
Therefore, by (24), we obtain the following corollary.
For with , we have
In particular, , we get
Let us assume that in (21). Then we have
Therefore, by Theorem 1, (21) and (25), we obtain the following theorem.
Theorem 6 For with , we have
Let () be Euler polynomials of order s. Then can be expressed as a linear combination of .
Assume that with .
By (6), we get
Therefore, by Theorem 1 and (26), we obtain the following theorem.
Theorem 7 For with , we have
For with , we have
By Theorem 1 and (27), we obtain the following theorem.
Theorem 8 For with , we have
Remarks (a) For , by (40) we get
Thus, for , we have
where is the Stirling number of the second kind.
Applying on both sides (), we get
From (28) and (30), we have
Remark Let us define two operators d, as follows:
From (31), we note that
Thus, by (31) and (32), we get
3 Further remarks
For any , forms a basis for . Let . Let . Then can be expressed as a linear combination of as follows:
Thus, by (6) and (35), we get
Now, for each , by (36) we get
Let us take in (37). Then, by (28) and (37), we get
Case 1. For , we have
Case 2. Let .
For , we have(40)
For , we have(41)
Thus, by (38), (39), (40) and (41), we can determine .
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This paper is supported in part by the Research Grant of Kwangwoon University in 2013.
The authors declare that they have no competing interests.
Both authors contributed equally to the manuscript and typed, read, and approved the final manuscript.