, 2013:103

Higher-order Bernoulli, Euler and Hermite polynomials

• Dae San Kim
• Taekyun Kim
• Dmitry V Dolgy
• Seog-Hoon Rim
Open Access
Review

Abstract

In (Kim and Kim in J. Inequal. Appl. 2013:111, 2013; Kim and Kim in Integral Transforms Spec. Funct., 2013, doi:10.1080/10652469.2012.754756), we have investigated some properties of higher-order Bernoulli and Euler polynomial bases in . In this paper, we derive some interesting identities of higher-order Bernoulli and Euler polynomials arising from the properties of those bases for .

Keywords

Vector Space Ordinary Differential Equation Linear Operator Functional Equation Explicit Expression

1 Introduction

For , let us define the Bernoulli polynomials of order r as follows:
(1)
In the special case, , are called the n th Bernoulli numbers of order r. As is well known, the Euler polynomials of order r are defined by the generating function to be
(2)
For , the Frobenius-Euler polynomials of order r are also given by
(3)
The Hermite polynomials are defined by the generating function to be:
(4)
Thus, by (4), we get
(5)
where are called the n th Hermite numbers. Let . Then is an -dimensional vector space over ℚ. In [8, 10], it is called that and are bases for . Let Ω denote the space of real-valued differential functions on . We define four linear operators on Ω as follows:
(6)
(7)
Thus, by (6) and (7), we get
(8)

where , .

In this paper, we derive some new interesting identities of higher-order Bernoulli, Euler and Hermite polynomials arising from the properties of bases of higher-order Bernoulli and Euler polynomials for .

2 Some identities of higher-order Bernoulli and Euler polynomials

First, we introduce the following theorems, which are important in deriving our results in this paper.

Theorem 1 [8]

For , let . Then we have

Theorem 2 [10]

For , let :
1. (a)
If , then we have

2. (b)
If , then

Let us take .

Then, by (5), we get
(9)
From Theorem 1 and (9), we can derive the following equation (10):
(10)

Therefore, by (10), we obtain the following theorem.

Theorem 3 For , we have
We recall an explicit expression for Hermite polynomials as follows:
(11)
By (11), we get
(12)

Thus, by Theorem 3 and (12), we obtain the following corollary.

Corollary 4 For , we have

Now, we consider the identities of Hermite polynomials arising from the property of the basis of higher-order Bernoulli polynomials in .

For , by (6) and (8), we get
(13)

Therefore, by Theorem 2 and (13), we obtain the following theorem.

Theorem 5 For , with , we have
Let us assume that , with . Then, by (b) of Theorem 2, we get
(14)

Therefore, by (14), we obtain the following theorem.

Theorem 6 For , with , we have
Remark From (12), we note that
(15)
and
(16)

Theorem 7 [10]

For , with and , we have

where is the Stirling number of the second kind and .

Theorem 8 [10]

For , with and , we have

Let us take . Then, by Theorem 7 and Theorem 8, we obtain the following corollary.

Corollary 9 For :
1. (a)
For , we have

2. (b)
For , we have

Theorem 10 [9]

For , we have
Let us take . Then
(17)

Therefore, by (17), we obtain the following corollary.

Corollary 11 For , we have
For , the Frobenius-Euler polynomials are defined by the generating function to be
(18)
Thus, by (18), we get
(19)
For , let . Then we note that
(20)
Let us take . Then, by (20), we get
(21)

Therefore, by (21), we obtain the following theorem.

Theorem 12 For , we have

Let us take on the both sides of Theorem 12.

Then, we have
(22)
By (22), we get
(23)

Notes

Acknowledgements

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

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Authors and Affiliations

• Dae San Kim
• 1
• Taekyun Kim
• 2
• Dmitry V Dolgy
• 3
• Seog-Hoon Rim
• 4
1. 1.Department of MathematicsSogang UniversitySeoulS. Korea
2. 2.Department of MathematicsKwangwoon UniversitySeoulS. Korea
3. 3.Hanrimwon, Kwangwoon UniversitySeoulS. Korea
4. 4.Department of Mathematics EducationKyungpook National UniversityTaeguS. Korea