Higher-order Bernoulli, Euler and Hermite polynomials
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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 .
KeywordsVector Space Ordinary Differential Equation Linear Operator Functional Equation Explicit Expression
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 
Theorem 2 
- (a)If , then we have
- (b)If , then
Let us take .
Therefore, by (10), we obtain the following theorem.
Thus, by Theorem 3 and (12), we obtain the following corollary.
Now, we consider the identities of Hermite polynomials arising from the property of the basis of higher-order Bernoulli polynomials in .
Therefore, by Theorem 2 and (13), we obtain the following theorem.
Therefore, by (14), we obtain the following theorem.
Theorem 7 
where is the Stirling number of the second kind and .
Theorem 8 
Let us take . Then, by Theorem 7 and Theorem 8, we obtain the following corollary.
- (a)For , we have
- (b)For , we have
Theorem 10 
Therefore, by (17), we obtain the following corollary.
Therefore, by (21), we obtain the following theorem.
Let us take on the both sides of Theorem 12.
This paper is supported in part by the Research Grant of Kwangwoon University in 2013.
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