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Barnes’ multiple Bernoulli and generalized Barnes’ multiple Frobenius-Euler mixed-type polynomials
Advances in Difference Equations volume 2014, Article number: 238 (2014)
Abstract
In this paper, by considering Barnes’ multiple Bernoulli polynomials as well as generalized Barnes’ multiple Frobenius-Euler polynomials, we define and investigate the mixed-type polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.
MSC:05A15, 05A40, 11B68, 11B75, 33E20, 65Q05.
1 Introduction
In this paper, we consider the polynomials
called Barnes’ multiple Bernoulli and generalized Barnes’ multiple Frobenius-Euler mixed-type polynomials, whose generating function is given by
where with , . When ,
are called Barnes’ multiple Bernoulli and generalized Barnes’ multiple Frobenius-Euler mixed-type numbers.
Recall that Barnes’ multiple Bernoulli polynomials, denoted by , are given by the generating function as
where [1, 2]. In addition, the generalized Barnes’ multiple Frobenius-Euler polynomials, denoted by , are given by the generating function as
are called Barnes-type Frobenius-Euler polynomials. If further and , then are called Frobenius-Euler polynomials of order s (see e.g. [8, 9]). If , then are called generalized Barnes-type Euler polynomials. These polynomials arise naturally in connection with the study of Barnes-type Peters polynomials. Peters polynomials were mentioned in [[10], p.128] and were investigated in e.g. [11].
In this paper, by considering Barnes’ multiple Bernoulli polynomials as well as generalized Barnes’ multiple Frobenius-Euler polynomials, we define and investigate the mixed-type polynomials of these polynomials. From the properties of Sheffer sequences of these polynomials arising from umbral calculus, we derive new and interesting identities.
2 Umbral calculus
Let ℂ be a complex number field and let ℱ be the set of all formal power series in the variable t:
Let and let be the vector space of all linear functionals on ℙ. is the action of the linear functional L on the polynomial , and we recall that the vector space operations on are defined by , , where c is a complex constant in ℂ. For , let us define the linear functional on ℙ by setting
In particular,
where is the Kronecker symbol.
For , we have . That is, . The map is a vector space isomorphism from onto ℱ. Henceforth, 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 will be thought of as both a formal power series and a linear functional. We call ℱ umbral algebra, and umbral calculus is the study of umbral algebra. The order of a power series (≠0) 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. For with and , there exists a unique sequence () such that for [[10], Theorem 2.3.1]. Such a sequence is called the Sheffer sequence for which is denoted by .
For and , we have
and
[[10], Theorem 2.2.5]. Thus, by (8), we get
Sheffer sequences are characterized in the generating function [[10], Theorem 2.3.4].
Lemma 1 The sequence is Sheffer for if and only if
where is the compositional inverse of .
For , we have the following equations [[10], Theorem 2.3.7, Theorem 2.3.5, Theorem 2.3.9]:
where .
Assume that and . Then the transfer formula [[10], Corollary 3.8.2] is given by
For and , assume that
Then we have [[10], p.132]
3 Main results
From definitions (2), (3) and (1), , and are the Appell sequences for
respectively. So,
In particular,
where , , and .
3.1 Explicit expressions
Let () with . The (signed) Stirling numbers of the first kind are defined by
Notice that
Theorem 1 We have
Proof By (11) with (16), we get
Thus, we obtain identity (18).
Next,
Thus, we obtain (19).
Finally, we obtain that
Thus, we get identity (20). □
3.2 The Sheffer identity
Theorem 2
Proof By (16) with
using (12), we have (21). □
3.3 Recurrence
Theorem 3
Remark When , as the left-hand side of (22) is equal to 0, we have
Proof By applying
[[10], Corollary 3.7.2] with (16), we get
Since
and
is a series with order at least one, we have
Here and (). Therefore, we obtain
which is (22). □
3.4 More relations
Theorem 4 For , we have
Proof For , we have
The third term is
Since
the first term is
Since
the second term is
Therefore, we obtain
which is identity (24). □
3.5 A relation including Bernoulli numbers
Theorem 5 For , we have
Proof We shall compute
in two different ways. On the one hand, it is equal to
On the other hand, it is equal to
The third term of (26) is equal to
The second term of (26) is equal to
Since
the first term of (26) is equal to
Therefore, we get, for ,
Dividing both sides by , we obtain, for ,
Thus, we get (25). □
3.6 A relation with Stirling numbers
The Stirling numbers of the second kind are defined by
Then
Theorem 6
Proof For (16) and (27), assume that . By (13), we have
Thus, we get identity (28). □
3.7 A relation with falling factorials
Theorem 7
Proof For (16) and (17), assume that . By (13), we have
Thus, we get identity (29). □
3.8 A relation with higher-order Frobenius-Euler polynomials
Theorem 8
Proof For (16) and
assume that . By (13), similarly to the proof of (25), we have
Thus, we get identity (30). □
3.9 A relation with higher-order Bernoulli polynomials
Bernoulli polynomials of order p are defined by
(see e.g. [[10], Section 2.2]).
Theorem 9
Proof For (16) and
assume that . By (13), similarly to the proof of (25), we have
Thus, we get identity (32). □
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Acknowledgements
The authors would like to thank the referees for their valuable comments. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MOE) (No. 2012R1A1A2003786) and was partially supported by Kwangwoon University in 2014.
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Kim, D.S., Kim, T., Komatsu, T. et al. Barnes’ multiple Bernoulli and generalized Barnes’ multiple Frobenius-Euler mixed-type polynomials. Adv Differ Equ 2014, 238 (2014). https://doi.org/10.1186/1687-1847-2014-238
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DOI: https://doi.org/10.1186/1687-1847-2014-238
Keywords
- Formal Power Series
- Linear Functional
- Bernoulli Polynomial
- Stirling Number
- Euler Polynomial