Abstract
In this paper, we derive the identities of higher-order Bernoulli, Euler and Frobenius-Euler polynomials from the orthogonality of Hermite polynomials. Finally, we give some interesting and new identities of several special polynomials arising from umbral calculus.
MSC: 05A10, 05A19.
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1 Introduction
The Hermite polynomials are defined by the generating function to be
with the usual convention about replacing by (see [1]). In the special case, , are called the nth Hermite numbers. From (1.1) we have
Thus, by (1.2), we get
where .
As is well known, the Bernoulli polynomials of order r are defined by the generating function to be
In the special case, , are called the n th Bernoulli numbers of order r (see [1–4]).
The Euler polynomials of order r are also defined by the generating function to be
In the special case, , are called the nth Euler numbers of order r.
For , the Frobenius-Euler polynomials of order r are given by
In the special case, , are called the nth Frobenius-Euler numbers of order r (see [1–16]).
The Stirling numbers of the first kind are defined by the generating function to be
and the Stirling numbers of the second kind are given by
In [1] it is known that from an orthogonal basis for the space
with respect to the inner product
For , let us assume that
Then, from the orthogonality of Hermite polynomials and Rodrigues’ formula, we have
In particular, for (), we easily get
Let ℱ be the set of all formal power series in the variable t over ℂ with
Let us assume that ℙ is the algebra of polynomials in the variable x over ℂ and that is the vector space of all linear functionals on ℙ. denotes the action of the linear functional L on polynomials , and we remind that the vector space structure on is defined by
where c is a complex constant (see [2, 11, 14]).
The formal power series
defines a linear functional on ℙ by setting
Thus, by (1.15) and (1.16), we get
where is the Kronecker symbol (see [2, 11, 14]).
Let . By (1.16), we get
Thus, by (1.18), we see that . The map is a vector space isomorphism from onto ℱ. Henceforth, ℱ will be thought of as both a formal power series and a linear functional. We call ℱ the umbral algebra. The umbral calculus is the study of umbral algebra (see [2, 11, 14]).
The order of the nonzero power series is the smallest integer k for which the coefficient of does not vanish. A series having is called a delta series, and a series having is called an invertible series (see [2, 11, 14]). By (1.16) and (1.17), we see that . For and , we have
Let and . Then we easily see that
From (1.19), we can derive the following equation:
Thus, by (1.21), we get
Let be a delta series, and let be an invertible series. Then there exists a unique sequence of polynomials with , where (see [2, 11, 14]). The sequence is called Sheffer sequence for , which is denoted by . For and , we have
and
In this paper, we introduce the identities of several special polynomials which are derived from the orthogonality of Hermite polynomials. Finally, we give some new and interesting identities of the higher-order Bernoulli, Euler and Frobenius-Euler polynomials arising from umbral calculus.
2 Some identities of several special polynomials
From (1.5), we note that
By (2.1), we get
From (1.5) and (2.2), we have
By (1.8) and (1.9), we get
Therefore, by (2.3) and (2.4), we obtain the following theorem.
Theorem 2.1 For , we have
By (1.5), we easily see that
Therefore, by Theorem 2.1 and (2.5), we obtain the following corollary.
Corollary 2.2 For , we have
Let us take . Then, by (1.11), we get
From (1.12), we can derive the computation of as follows:
where
From (2.7) and (2.8), we can derive the following equation:
Therefore, by Corollary 2.2, (2.6) and (2.9), we obtain the following theorem.
Theorem 2.3 For , we have
By (1.4), we easily see that
Thus, by (2.10), we get
From (1.4) and (2.11), we have
By (1.19), we easily get
From (1.8), (1.21) and (2.13), we have
Thus, by (2.12) and (2.14), we get
Therefore, by (2.12) and (2.15), we obtain the following theorem.
Theorem 2.4 For , we have
By (1.4), we easily get
Therefore, by Theorem 2.4 and (2.16), we obtain the following corollary.
Corollary 2.5 For , we have
Let us consider . Then, by (1.11), can be written as
Now, we compute ’s for as follows:
where
By Corollary 2.5 and (2.19), we get
From (2.18) and (2.20), we have
Therefore, by (2.17) and (2.21), we obtain the following theorem.
Theorem 2.6 For , we have
It is easy to show that
From (1.6) and (2.22), we have
where
Thus, by (2.24), we get
From (2.23) and (2.25), we can derive the following equation:
By (1.6), we easily see that
Therefore, by (2.26) and (2.27), we obtain the following theorem.
Theorem 2.7 For , we have
Let us take . Then, by (1.11), is given by
By (1.12), we get
where
By (2.29) and (2.30), we get
Therefore, by (2.28) and (2.31), we obtain the following theorem.
Corollary 2.8 For , we have
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Acknowledgements
The authors would like to express their sincere gratitude to referees for their valuable comments. 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. Some new identities of Bernoulli, Euler and Hermite polynomials arising from umbral calculus. Adv Differ Equ 2013, 73 (2013). https://doi.org/10.1186/1687-1847-2013-73
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DOI: https://doi.org/10.1186/1687-1847-2013-73