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Sheffer sequences of polynomials and their applications
Advances in Difference Equations volume 2013, Article number: 118 (2013)
Abstract
In this paper, we investigate some properties of several Sheffer sequences of several polynomials arising from umbral calculus. From our investigation, we can derive many interesting identities of several polynomials.
MSC:05A40, 05A19.
1 Introduction
As is well known, the Bernoulli polynomials of order a are defined by the generating function to be
and the Narumi polynomials are also given by
In the special case, , are called the Narurni numbers.
Throughout this paper, we assume that with . Frobenius-Euler polynomials of order a are defined by the generating function to be
The Stirling number of the second kind is also defined by the generating function to be
and the Stirling number of the first kind is given by
Let
Let ℙ be the algebra of polynomials in the variable x over ℂ and be the vector space of all linear functionals on ℙ. The action of the linear functional L on a polynomial is denoted by . We recall that the vector space structures on are defined by , , where c is a complex constant (see [11, 12]).
For , we define a linear functional on ℙ by setting
By (1.6) and (1.7), we get
where is the Kronecker symbol (see [9–13]).
Suppose that . Then we have and . Thus, we note 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 shall call ℱ the umbral algebra. The umbral calculus is the study of umbral algebra (see [9–13]).
The order of the non-zero 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. Let and . Then there exists a unique sequence of polynomials such that (). The sequence is called Sheffer sequence for , which is denoted by . By (1.8), we easily get that . For and , we have
and
where (see [9–12]). For and , by (1.9), we get
Thus, by (1.11), we have
Let . Then we have
where is the compositional inverse of (see [11, 12]). By (1.2) and (1.13), we see that .
For , the Poisson-Charlier sequences are given by
In particular, , we have
The Frobenius-type Eulerian polynomials of order a are given by
From (1.13) and (1.16), we note that
Let us assume that , . Then we have
Equation (1.17) is important in deriving our results in this paper. The purpose of this paper is to investigate some properties of Sheffer sequences of several polynomials arising from umbral calculus. From our investigation, we can derive many interesting identities of several polynomials.
2 Sheffer sequences of polynomials
Let us assume that . Then, by the definition of Sheffer sequence, we see that . If is an invertible series, then is also an invertible series. Let us consider the following Sheffer sequences:
From (1.17) and (2.1), we note that
For , by (2.2), we get
Therefore, by (2.3), we obtain the following theorem.
Theorem 2.1 For and , we have
For example, let , where is the n th Daehee polynomial (see [1, 8, 9]). Then, by Theorem 2.1, we get
Let us take (). Then, by Theorem 2.1, we get
Therefore, by (2.4), we obtain the following theorem.
Theorem 2.2 For , let , . Then we have
Let
From Theorem 2.1, we can derive
Therefore, by (2.6), we obtain the following theorem.
Theorem 2.3 For , let , . Then we have
Let us take the following Sheffer sequence:
By Theorem 2.1 and (2.7), we get
where are the n th Euler polynomials of order α which is defined by the generating function to be
Therefore, by (2.8), we obtain the following theorem.
Theorem 2.4 For , let . Then we have
As is known, we note that
Thus, by Theorem 2.1 and (2.9), we get
Therefore, by Theorem 2.4 and (2.10), we obtain the following corollary.
Corollary 2.5 For , and , we have
Remark Let . Then, by Theorem 2.1, we get
Let us assume that
Then, by Theorem 2.1 and (2.12), we get
Therefore, by (2.13), we obtain the following theorem.
Theorem 2.6 For , let , . Then we have
As is well known, the Bernoulli polynomials of the second kind are defined by the generating function to be
Thus, by (1.10) and (2.14), we get
By Theorem 2.1, (2.12) and (2.15), we get
Therefore, by Theorem 2.6 and (2.16), we obtain the following theorem.
Theorem 2.7 For , , we have
Remark From (1.2), we note that
where . By Theorem 2.1, (2.12) and (2.17), we get
From (2.16) and (2.18), we can derive the following identity:
where , and . Let
From Theorem 2.1 and (2.20), we note that
Therefore, by (2.21), we obtain the following proposition.
Proposition 2.8 For , let , . Then we have
Now we observe that
By Theorem 2.1, (2.20) and (2.22), we get
Therefore, by Proposition 2.8 and (2.23), we obtain the following theorem.
Theorem 2.9 For , and , we have
Remark
It is easy to show that
By Theorem 2.1, (2.7) and (2.24), we get
From Theorem 2.4 and (2.25), we can derive the following identity:
Let us consider the following Sheffer sequence:
By Theorem 2.1 and (2.27), we get
From (1.15) and (2.28), we can derive
Therefore, by (2.29), we obtain the following theorem.
Theorem 2.10 For , let , where , and . Then 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., Rim, SH. et al. Sheffer sequences of polynomials and their applications. Adv Differ Equ 2013, 118 (2013). https://doi.org/10.1186/1687-1847-2013-118
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DOI: https://doi.org/10.1186/1687-1847-2013-118