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Some new identities of Frobenius-Euler numbers and polynomials
Journal of Inequalities and Applications volume 2012, Article number: 307 (2012)
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.
Let . As is well known, the Frobienius-Euler polynomials are defined by the generating function to be
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
By (14) and (15), we get
Therefore, by (16), we obtain the following theorem.
Theorem 2 For , we have
From Theorem 2, we note that can be generated by as follows:
By (17), we get
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
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
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
Let us take . Then, by Theorem 5, can be generated by as follows:
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
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
Continuing this process, we obtain the following equation:
By (1), (2) and (49), we get
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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.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
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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
- Positive Integer
- Linear Combination
- Vector Space
- Good Basis
- Monic Polynomial