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Higher-order Bernoulli, Euler and Hermite polynomials
Advances in Difference Equations volume 2013, Article number: 103 (2013)
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 .
For , let us define the Bernoulli polynomials of order r as follows:
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
For , the Frobenius-Euler polynomials of order r are also given by
The Hermite polynomials are defined by the generating function to be:
Thus, by (4), we get
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:
Thus, by (6) and (7), we get
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 
For , let . Then we have
Theorem 2 
For , let :
If , then we have
If , then
Let us take .
Then, by (5), we get
From Theorem 1 and (9), we can derive the following equation (10):
Therefore, by (10), we obtain the following theorem.
Theorem 3 For , we have
We recall an explicit expression for Hermite polynomials as follows:
By (11), we get
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
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
Therefore, by (14), we obtain the following theorem.
Theorem 6 For , with , we have
Remark From (12), we note that
Theorem 7 
For , with and , we have
where is the Stirling number of the second kind and .
Theorem 8 
For , with and , we have
Let us take . Then, by Theorem 7 and Theorem 8, we obtain the following corollary.
Corollary 9 For :
For , we have
For , we have
Theorem 10 
For , we have
Let us take . Then
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
Thus, by (18), we get
For , let . Then we note that
Let us take . Then, by (20), we get
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
By (22), we get
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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.
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.