# A note on higher-order Bernoulli polynomials

Open Access
Research

## Abstract

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.

### Keywords

Linear Combination Vector Space Linear Operator Dimensional Vector Good Basis

## 1 Introduction

Let and . For a fixed , the n th Bernoulli polynomials are defined by the generating function to be
(1)

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
(2)
Thus, by (2) we get the Euler-type sums of products of Bernoulli numbers as follows:
(3)

By (2) and (3), we see that is a monic polynomial of degree n with coefficients in Q.

From (2), we note that
(4)
and
(5)
Let Ω denote the space of real-valued differential functions on . Now, we define three linear operators I, △, D on Ω as follows:
(6)

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:
(7)
From (6) and (7), we can derive the following identities:
(8)
By (5) and (8), we get
(9)
It is easy to show that
(10)
and
(11)
By (7) and (9), we get
(12)
From (6) and (12), we note that
(13)
Thus, by (13) we get
(14)
Hence, from (14) we have
(15)

Case 1. Let . Then for all .

By (15), we get
(16)
Case 2. Assume that .
1. (i)
For , by (15) we get
(17)

2. (ii)
For , by (15) we see that
(18)

Therefore, by (7), (16), (17) and (18), we obtain the following theorem.

Theorem 1
1. (a)
For , we have

2. (b)
For , we have

Let us take . Then can be expressed as a linear combination of . For , we have
(19)

Therefore, by Theorem 1 and (19), we obtain the following corollary.

Corollary 2 For with , we have
Let us assume that with . Observe that
(20)

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:
(21)
For , we have
(22)

Thus, by Theorem 1 and (22), we obtain the following theorem.

Theorem 4 For with , we have
In particular, for , we have
(23)
By comparing coefficients on the both sides of (23), we get
(24)

Therefore, by (24), we obtain the following corollary.

Corollary 5
1. (a)
For with , we have

2. (b)
In particular, , we get

Let us assume that in (21). Then we have
(25)

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
(26)

Therefore, by Theorem 1 and (26), we obtain the following theorem.

Theorem 7 For with , we have
For with , we have
(27)

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
(28)
where is the Stirling number of the second kind.
1. (b)
Assume
(29)

Applying on both sides (), we get
(30)
From (28) and (30), we have
Remark Let us define two operators d, as follows:
(31)
From (31), we note that
(32)
Thus, by (31) and (32), we get
(33)
and
(34)

## 3 Further remarks

For any , forms a basis for . Let . Let . Then can be expressed as a linear combination of as follows:
(35)
Thus, by (6) and (35), we get
(36)
Now, for each , by (36) we get
(37)
Let us take in (37). Then, by (28) and (37), we get
(38)
Case 1. For , we have
(39)
Case 2. Let .
1. (i)
For , we have
(40)

2. (ii)
For , we have
(41)

Thus, by (38), (39), (40) and (41), we can determine .

## Notes

### Acknowledgements

This paper is supported in part by the Research Grant of Kwangwoon University in 2013.

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