Abstract
In this paper, we consider higher-order Frobenius-Euler polynomials, associated with poly-Bernoulli polynomials, which are derived from polylogarithmic function. These polynomials are called higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials. The purpose of this paper is to give various identities of those polynomials arising from umbral calculus.
Similar content being viewed by others
1 Introduction
For with , the Frobenius-Euler polynomials of order α () are defined by the generating function to be
When , are called the Frobenius-Euler numbers of order α. As is well known, the Bernoulli polynomials of order α are defined by the generating function to be
When , is called the n th Bernoulli number of order α. In the special case, , is called the n th Bernoulli polynomial. When , is called the n th ordinary Bernoulli number. Finally, we recall that the Euler polynomials of order α are given by
When , is called the n th Euler number of order α. In the special case, , is called the n th ordinary Euler polynomial. The classical polylogarithmic function is defined by
As is known, poly-Bernoulli polynomials are defined by the generating function to be
Let ℂ be the complex number field, and let ℱ be the set of all formal power series in the variable t over ℂ with
Now, we use the notation . In this paper, will be denoted by the vector space of all linear functionals on ℙ. Let us assume that be the action of the linear functional L on the polynomial , and we remind that the vector space operations on are defined by , , where c is a complex constant in ℂ. The formal power series
defines a linear functional on ℙ by setting
From (1.7) and (1.8), we note that
where is the Kronecker symbol.
Let us consider . Then we see that , and so as linear functionals. The map is a vector space isomorphism from onto ℱ. Henceforth, ℱ will denote 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 (see [14]). We shall call ℱ the umbral algebra. The umbral calculus is the study of umbral algebra. The order of a nonzero power series is the smallest integer k, for which the coefficient of does not vanish. A series is called a delta series if , and an invertible series if . Let . Then we have
For with , , there exists a unique sequence () such that for . The sequence is called the Sheffer sequence for , which is denoted by (see [14, 15]). Let and . Then we have
From (1.11), we note that
By (1.12), we get
From (1.13), we easily derive the following equation
For , , it is known that
Let . Then we have
where is the compositional inverse of with , and
The Stirling number of the second kind is defined by the generating function to be
For , it is well known that
Let , . Then we have
where
In this paper, we study higher-order Frobeniuns-Euler polynomials associated with poly-Bernoulli polynomials, which are called higher-order Frobenius-Euler and poly-Beroulli mixed-type polynomials. The purpose of this paper is to give various identities of those polynomials arising from umbral calculus.
2 Higher-order Frobenius-Euler polynomials, associated poly-Bernoulli polynomials
Let us consider the polynomials , called higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials, as follows:
where with , .
When , is called the n th higher-order Frobenius-Euler and poly-Bernoulli mixed type number.
From (1.16) and (2.1), we note that
By (1.17) and (2.2), we get
From (2.1), we can easily derive the following equation
By (1.16) and (2.2), we get
In [7], it is known that
Thus, by (2.5) and (2.6), we get
By (1.1), we easily see that
Therefore, by (2.7) and (2.8), we obtain the following theorem.
Theorem 2.1 For , , we have
In [7], it is known that
By (2.5) and (2.9), we get
Therefore, by (2.8) and (2.10), we obtain the following theorem.
Theorem 2.2 For , , we have
From (1.19) and (2.2), we have
Now, we note that
By (2.11) and (2.12), we get
It is easy to show that
For any delta series , we have
Thus, by (2.13), (2.14) and (2.15), we get
Therefore, by (2.16), we obtain the following theorem.
Theorem 2.3 For , , we have
Remark 1 If , then we have
Thus, by (2.17), we get .
From (2.4), we have
Applying t on both sides of Theorem 2.3, we get
Thus, by (2.19), we have
Therefore, by (2.20), we obtain the following theorem.
Theorem 2.4 For , with , we have
From (1.14) and (2.5), we note that
By (1.15) and (2.21), we get
Therefore, by (2.22), we obtain the following theorem.
Theorem 2.5 For , , we have
Now, we compute in two different ways.
On the one hand,
On the other hand, we get
Therefore, by (2.23) and (2.24), we obtain the following theorem.
Theorem 2.6 For , , we have
Now, we consider the following two Sheffer sequences:
where , and with . Let us assume that
By (1.21) and (2.26), we get
Therefore, by (2.26) and (2.27), we obtain the following theorem.
Theorem 2.7 For , , we have
From (1.3) and (2.1), we note that
where , .
By the same method, we get
From (1.1) and (2.1), we note that
where , and with , , .
Let us assume that
By (1.21) and (2.31), we get
Therefore, by (2.31) and (2.32), we obtain the following theorem.
Theorem 2.8 For , , we have
It is known that
Let
Then, by (1.21) and (2.34), we get
Therefore, by (2.34) and (2.35), we obtain the following theorem.
Theorem 2.9 For , we have
Finally, we consider the following two Sheffer sequences:
where .
Let us assume that
Then, by (1.21) and (2.37), we get
Therefore, by (2.37) and (2.38), we obtain the following theorem.
Theorem 2.10 For , , we have
References
Araci S, Acikgoz M: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. 2012, 22(3):399–406.
Kim T: An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p -adic invariant q -integrals on . Rocky Mt. J. Math. 2011, 41(1):239–247. 10.1216/RMJ-2011-41-1-239
Kim T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory 2012, 132(1):2854–2865.
Ryoo C: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc. 2011, 14(4):495–501.
Ryoo CS, Agarwal RP: Exploring the multiple Changhee q -Bernoulli polynomials. Int. J. Comput. Math. 2005, 82(4):483–493. 10.1080/00207160512331323362
Kim DS, Kim T, Kim YH, Lee SH: Some arithmetic properties of Bernoulli and Euler numbers. Adv. Stud. Contemp. Math. 2012, 22(4):467–480.
Kim, DS, Kim, T: Poly-Bernoulli polynomials arising from umbral calculus (communicated)
Kim T: Power series and asymptotic series associated with the q -analog of the two-variable p -adic L -function. Russ. J. Math. Phys. 2005, 12(2):186–196.
Can M, Cenkci M, Kurt V, Simsek Y: Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius-Euler l -functions. Adv. Stud. Contemp. Math. 2009, 18(2):135–160.
Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20(1):7–21.
Kim T, Choi J: A note on the product of Frobenius-Euler polynomials arising from the p -adic integral on . Adv. Stud. Contemp. Math. 2012, 22(2):215–223.
Kurt B, Simsek Y: On the generalized Apostol-type Frobenius-Euler polynomials. Adv. Differ. Equ. 2013., 2013: Article ID 1
Simsek Y, Yurekli O, Kurt V: On interpolation functions of the twisted generalized Frobenius-Euler numbers. Adv. Stud. Contemp. Math. 2007, 15(2):187–194.
Roman S Pure and Applied Mathematics 111. In The Umbral Calculus. Academic Press, New York; 1984.
Roman S, Rota G-C: The umbral calculus. Adv. Math. 1978, 27(2):95–188. 10.1016/0001-8708(78)90087-7
Acknowledgements
This work was supported by the National Research Foundation of Korea (NRF) grant, funded by the Korea government (MOE) (No. 2012R1A1A2003786).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Kim, D.S., Kim, T. Higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials. Adv Differ Equ 2013, 251 (2013). https://doi.org/10.1186/1687-1847-2013-251
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2013-251