Abstract
In this paper, we give some new and interesting identities which are derived from the basis of Frobenius-Euler. Recently, several authors have studied some identities of Frobenius-Euler polynomials. From the methods of our paper, we can also derive many interesting identities of Frobenius-Euler numbers and polynomials.
Similar content being viewed by others
1 Introduction
Let . As is well known, the Frobienius-Euler polynomials are defined by the generating function to be
with the usual convention about replacing by (see [1–6]).
In the special case, , are called the n th Frobenius-Euler numbers.
Thus, by (1), we get
where is the Kronecker symbol.
From (1), we can derive the following equation:
Thus, by (3), we easily see that the leading coefficient of is . So, are monic polynomials of degree n with coefficients in .
From (1), we have
Thus, by (4), we get
It is easy to show that
From (6), we have
Let be a vector space over . Then we note that is a good basis for .
In this paper, we develop some new methods to obtain some new identities and properties of Frobenius-Euler polynomials which are derived from the basis of Frobenius-Euler polynomials. Those methods are useful in studying the identities of Frobenius-Euler polynomials.
2 Some identities of Frobenius-Euler polynomials
Let us take . Then can be expressed as a -linear combination of as follows:
Let us define the operator by
From (9), we can derive the following equation (10):
For , let us take the r th derivative of in (10) as follows:
Thus, by (11), we get
From (12), we have
where and . Therefore, by (13), we obtain the following theorem.
Theorem 1 For , , let with . Then we have
Let us take . Then, by Theorem 1, we get
where
By (14) and (15), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2012-307/MediaObjects/13660_2012_Article_435_Equ16_HTML.gif)
Therefore, by (16), we obtain the following theorem.
Theorem 2 For , we have
Let
From Theorem 2, we note that can be generated by as follows:
By (17), we get
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2012-307/MediaObjects/13660_2012_Article_435_Equ20_HTML.gif)
From (18) and (20), we have
Therefore, by (21), we obtain the following theorem.
Theorem 3 For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2012-307/MediaObjects/13660_2012_Article_435_Equc_HTML.gif)
Let us consider
By Theorem 1, can be expressed by
From (22), we have
By Theorem 1, we get
Therefore, by (25), we obtain the following theorem.
Theorem 4 For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1186%2F1029-242X-2012-307/MediaObjects/13660_2012_Article_435_Equd_HTML.gif)
3 Higher-order Frobenius-Euler polynomials
For , the Frobenius-Euler polynomials of order r are defined by the generating function to be
with the usual convention about replacing by (see [1–10]). In the special case, , are called the n th Frobenius-Euler numbers of order r (see [8, 9]).
From (26), we have
with the usual convention about replacing by .
By (26), we get
where . From (27) and (28), we note that the leading coefficient of is given by
Thus, by (29), we see that is a monic polynomial of degree n with coefficients in . From (26), we have
and
It is not difficult to show that
Now, we note that is also a good basis for .
Let us define the operator D as and let . Then can be written as
From (9) and (32), we have
Thus, by (33) and (34), we get
Let us take the k th derivative of in (35).
Then we have
Thus, from (36), we have
Thus, by (37), we get
Therefore, by (33) and (38), we obtain the following theorem.
Theorem 5 For , let with
Then we have
That is,
Let us take . Then, by Theorem 5, can be generated by as follows:
where
and
By (40) and (41), we get
Therefore, by (39) and (42), we obtain the following theorem.
Theorem 6 For , we have
Let us assume that .
Then we have
From Theorem 1, we note that can be expressed as a linear combination of
where
By (34) and (45), we get
Therefore, by (44) and (46), we obtain the following theorem.
Theorem 7 For , we have
Remark From (2) and (37), we note that
and
Continuing this process, we obtain the following equation:
By (1), (2) and (49), we get
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.
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.
Cangul IN, Kurt V, Ozden H, Simsek Y: On the higher-order w - q -Genocchi numbers. Adv. Stud. Contemp. Math. 2009, 19(1):39–57.
Carlitz L: A note on q -Eulerian numbers. J. Comb. Theory, Ser. A 1978, 25(1):90–94. 10.1016/0097-3165(78)90038-9
Kim DS, Dolgy DV, Kim T, Rim S-H: Some formulae for the product of two Bernoulli and Euler polynomials. Abstr. Appl. Anal. 2012., 2012: Article ID 784307
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.
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(12):2854–2865. 10.1016/j.jnt.2012.05.033
Kim T: Symmetry of power sum polynomials and multivariate fermionic p -adic invariant integral on . Russ. J. Math. Phys. 2009, 16(1):93–96. 10.1134/S1061920809010063
Rim S-H, Jeong J: On the modified q -Euler numbers of higher order with weight. Adv. Stud. Contemp. Math. 2012, 22(1):93–98.
Rim S-H, Lee J: Some identities on the twisted -Genocchi numbers and polynomials associated with q -Bernstein polynomials. Int. J. Math. Math. Sci. 2011., 2011: Article ID 482840
Ryoo CS: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc. 2011, 14(4):495–501.
Simsek Y, Bayad A, Lokesha V: q -Bernstein polynomials related to q -Frobenius-Euler polynomials, l -functions, and q -Stirling numbers. Math. Methods Appl. Sci. 2012, 35(8):877–884. 10.1002/mma.1580
Simsek Y, Yurekli O, Kurt V: On interpolation functions of the twisted generalized Frobenius-Euler numbers. Adv. Stud. Contemp. Math - Jang’jun Math. Soc. 2007, 15(2):187–194.
Shiratani K: On the Euler numbers. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 1973, 27: 1–5.
Shiratani K, Yamamoto S: On a p -adic interpolation function for the Euler numbers and its derivatives. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 1985, 39(1):113–125.
Acknowledgements
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology 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. Some new identities of Frobenius-Euler numbers and polynomials. J Inequal Appl 2012, 307 (2012). https://doi.org/10.1186/1029-242X-2012-307
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2012-307