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Some identities of Frobenius-Euler polynomials arising from umbral calculus
Advances in Difference Equations volume 2012, Article number: 196 (2012)
In this paper, we study some interesting identities of Frobenius-Euler polynomials arising from umbral calculus.
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
We use notation and denotes the vector space of all linear functional on ℙ.
Also, denotes the action of the linear functional L on the polynomial , and we remind that the vector space operations on is defined by
where c is any constant in C.
The formal power series
defines a linear functional on ℙ by setting
In addition, the map is a vector space isomorphism from onto F (see [1, 2]). 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. We shall call F the umbral algebra (see [1, 2]).
and for all polynomial
For , we have
where the sum is over all nonnegative integers such that (see [1, 2]). The order of the power series is the smallest integer k for which does not vanish. We define if . We see that and . The series has a multiplicative inverse, denoted by or , if and only if . Such series is called an invertible series. A series for which is called a delta series (see [1, 2]). For , we have .
A delta series has a compositional inverse such that .
For , we have .
From (5), we have
Thus, we see that
By (6), we get
By (7), we have
Let be a polynomial with .
Let be a delta series, and let be an invertible series. Then there exists a unique sequence of polynomials such that for all . The sequence is called the Sheffer sequence for or that is Sheffer for .
The Sheffer sequence for is called the associated sequence for or is associated to . The Sheffer sequence for is called the Appell sequence for or is Appell for (see [1, 2]). The umbral calculus is the study of umbral algebra and the modern classical umbral calculus can be described as a systemic study of the class of Sheffer sequences. Let . Then we have
Let be Sheffer for . Then
For , we recall that the Frobenius-Euler polynomials are defined by the generating function to be
with the usual convention about replacing by (see ). In the special case, , are called the n th Frobenius-Euler numbers. By (16), we get
From (17), we note that the leading coefficient of is . So, is a monic polynomial of degree n with coefficients in .
In this paper, we derive some new identities of Frobenius-Euler polynomials arising from umbral calculus.
2 Applications of umbral calculus to Frobenius-Euler polynomials
Let be an Appell sequence for . From (14), we have
For , let us take .
Then we see that is an invertible series.
From (16), we have
By (20), we get
and by (17), we get
Therefore, by (21) and (22), we obtain the following proposition.
Proposition 1 For , , we see that is the Appell sequence for .
From (20), we have
By (21) and (23), we get
Therefore, by (24) we obtain the following theorem.
Theorem 2 Let . Then we have
From (16), we have
By (25), we get
From Theorem 2, we can derive the following equation (27):
By (27), we get
From (8) and (28), we have
Therefore, by (26), we obtain the following theorem.
Theorem 3 For , we have
From (16), (17), and (18), we note that
Therefore, by (29), we obtain the following theorem.
Theorem 4 For , , we have
By (15) and Proposition 1, we get
From (30), we can derive equation (31):
By (11) and (31), we get
Therefore, by (32), we obtain the following corollary.
Corollary 5 For , we have
Let be a vector space over .
For , let us take
By Proposition 1, is an Appell sequence for where . Thus, we have
From (33) and (34), we can derive
Thus, by (35), we get
From (11) and (36), we have
Therefore, by (37), we obtain the following theorem.
Theorem 6 For , let us assume that . Then we have
The higher-order Frobenius-Euler polynomials are defined by
In the special case, , are called the n th Frobenius-Euler numbers of order r. From (38), we have
Note that is a monic polynomial of degree n with coefficients in .
For , , let . Then we easily see that is an invertible series.
From (38) and (39), we have
By (40), we get
Therefore, by (41) and (42), we obtain the following proposition.
Proposition 7 For , is an Appell sequence for
Remark Note that
From (43), we have
By (43), (44), and (45), we get
Let us take with
From the definition of Appell sequences, we have
By (46) and (47), we get
Thus, from (48), we have
Therefore, by (46) and (49), we obtain the following theorem.
Theorem 8 For , let
Then we have
where and .
Remark Let be a Sheffer sequence for . Then Sheffer identity is given by
From (21), Proposition 1, and (50), we have
By Proposition 7 and (50), we get
Let . Then we have
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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 identities of Frobenius-Euler polynomials arising from umbral calculus. Adv Differ Equ 2012, 196 (2012). https://doi.org/10.1186/1687-1847-2012-196
- Vector Space
- Formal Power Series
- Linear Functional
- Monic Polynomial
- Multiplicative Inverse