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Higher-order Bernoulli, Euler and Hermite polynomials
Advances in Difference Equations volume 2013, Article number: 103 (2013)
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 .
1 Introduction
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 [8]
For , let . Then we have
Theorem 2 [10]
For , let :
-
(a)
If , then we have
-
(b)
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
and
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 :
-
(a)
For , we have
-
(b)
For , we have
Theorem 10 [9]
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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Acknowledgements
This paper is supported in part by the Research Grant of Kwangwoon University in 2013.
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Kim, D.S., Kim, T., Dolgy, D.V. et al. Higher-order Bernoulli, Euler and Hermite polynomials. Adv Differ Equ 2013, 103 (2013). https://doi.org/10.1186/1687-1847-2013-103
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DOI: https://doi.org/10.1186/1687-1847-2013-103