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.
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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
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
From (18) and (20), we have
Therefore, by (21), we obtain the following theorem.
Theorem 3 For , we have
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
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
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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.
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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
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DOI: https://doi.org/10.1186/1029-242X-2012-307