- Open Access
Some identities of Frobenius-Euler polynomials arising from umbral calculus
© Kim and Kim; licensee Springer 2012
Received: 18 October 2012
Accepted: 5 November 2012
Published: 14 November 2012
In this paper, we study some interesting identities of Frobenius-Euler polynomials arising from umbral calculus.
We use notation and denotes the vector space of all linear functional on ℙ.
where c is any constant in C.
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]).
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 .
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 .
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
For , let us take .
Then we see that is an invertible series.
Therefore, by (21) and (22), we obtain the following proposition.
Proposition 1 For , , we see that is the Appell sequence for .
Therefore, by (24) we obtain the following theorem.
Therefore, by (26), we obtain the following theorem.
Therefore, by (29), we obtain the following theorem.
Therefore, by (32), we obtain the following corollary.
Let be a vector space over .
Therefore, by (37), we obtain the following theorem.
Note that is a monic polynomial of degree n with coefficients in .
For , , let . Then we easily see that is an invertible series.
Therefore, by (41) and (42), we obtain the following proposition.
Therefore, by (46) and (49), we obtain the following theorem.
where and .
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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