Abstract
In this paper, we derive some identities of Bernoulli, Euler, and Abel polynomials arising from umbral calculus.
MSC:05A10, 05A19.
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1 Introduction
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 is the vector space of all linear functionals on ℙ. denotes the action of the linear functional L on a polynomial , and we remind that the vector space structure on is defined by
where c is a complex constant (see [1–4]).
The formal power series
defines a linear functional on ℙ by setting
Thus, by (1.2) and (1.3), we get
where is the Kronecker symbol (see [3]).
For , from (1.4), we have
By (1.5), we get . The map is a vector space isomorphism from onto ℱ. So, ℱ denotes both the algebra of formal power series in t and the vector space of all linear functionals on ℙ, and so an element of ℱ is thought of as both a formal power series and a linear functional (see [1–3]). We call ℱ the umbral algebra, and the study of umbral algebra is called umbral calculus (see [1–3]).
The order of the nonzero power series is the smallest integer k for which the coefficient of does not vanish. If , then is called a delta series. If , then is called an invertible series (see [3]).
Let be polynomials in the variable x with degree n, and let and . Then there exists a unique sequence such that , where . The sequence is called the Sheffer sequence for , which is denoted by (see [3]).
For , we have
and
By (1.6), we get
Thus, from (1.8), we have
For , the following equations from (1.10) to (1.14) are well known in [3]:
where is the compositional inverse of , and
where .
The Euler polynomials of order r are defined by the generating function to be
with the usual convention about replacing by . In the special case, , are called the Euler numbers of order r.
As is well known, the higher-order Bernoulli polynomials are also defined by the generating function to be
with the usual convention about replacing by . In the special case, , are called the Bernoulli numbers of order r.
Recently, several researchers have studied the umbral calculus related to special polynomials. In this paper, we derive some interesting identities related to Bernoulli, Euler, and Abel polynomials arising from umbral calculus.
2 Some identities of special polynomials
It is known [3] that
where and . From (2.1), we have
where is the Stirling number of the second kind. Therefore, by (2.2), we obtain the following theorem.
Theorem 2.1 For and , we have
where .
In [3], we note that
and
where .
For , we have
The Bernoulli polynomials of the second kind are defined by the generating function to be
By (2.5) and (2.6), we get
Therefore, by (2.4) and (2.7), we obtain the following theorem.
Theorem 2.2 For , with , we have
It is well known (see [3]) that
Thus, by (2.8), we get
and
Therefore, by (2.9) and (2.10), we obtain the following lemma.
Lemma 2.3 For , we have
Let us consider the following sequences:
Then from (2.11), we have
Therefore, by (2.12), we obtain the following proposition.
Proposition 2.4 For , , we have
The Abel sequence is given by
By Proposition 2.4 and (2.13), we get
Therefore, by (2.14), we obtain the following theorem.
Theorem 2.5 For and , we have
Let us consider the following Sheffer sequences:
By (2.15), we note that
For , from (2.15), we have
and
From (2.18), we can derive the following equation (2.19):
Thus, by (2.19), we get
From (2.16), (2.17), and (2.20), we can derive the following equation (2.21):
and
where is the Stirling number of the first kind. By (2.22), we get
Thus, by (2.21) and (2.23), we get
From (1.14), we have
Therefore, by (2.24) and (2.25), we obtain the following theorem.
Theorem 2.6 For , , , we have
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Acknowledgements
The authors express their sincere gratitude to the referees for their valuable suggestions and 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., Lee, SH. et al. Some identities of Bernoulli, Euler and Abel polynomials arising from umbral calculus. Adv Differ Equ 2013, 15 (2013). https://doi.org/10.1186/1687-1847-2013-15
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DOI: https://doi.org/10.1186/1687-1847-2013-15