- Open Access
Frobenius-Euler polynomials and umbral calculus in the p-adic case
© Kim et al.; licensee Springer 2012
Received: 21 November 2012
Accepted: 6 December 2012
Published: 20 December 2012
In this paper, we study some p-adic Frobenius-Euler measure related to umbral calculus in the p-adic case. Finally, we derive some identities of Frobenius-Euler polynomials from our study.
Let p be a fixed prime number. Throughout this paper , and will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of , respectively.
If g is a function on , we denote by the same g the function on X. Namely, we can consider g as a function on X.
where the p-adic absolute value on is normalized by .
where is the Kronecker symbol.
Let and denote the vector space of all linear functionals on ℙ.
Here, ℱ denotes 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 (see [11, 15]). We will call ℱ the umbral algebra. The umbral calculus is the study of umbral algebra (see [11, 15]).
Let denote a polynomial of degree n. Suppose that with and . Then there exists a unique sequence of polynomials satisfying for all . The sequence is called the Sheffer sequence for , which is denoted by . If , then is called the Appell sequence for (see ).
where is compositional inverse of (see [11, 15]). In , Kim and Kim have studied some identities of Frobenius-Euler polynomials arising from umbral calculus. In this paper, we study some p-adic Frobenius-Euler integral on related to umbral calculus in the p-adic case. Finally, we derive some new and interesting identities of Frobenius-Euler polynomials from our study.
2 Frobenius-Euler polynomials associated with umbral calculus
Therefore, by (2.15), we obtain the following theorem.
Therefore, by (2.17), we obtain the following theorem.
Therefore, by (2.30), we obtain the following theorem.
Therefore, by (2.31) and (2.32), we obtain the following theorem.
where k is a positive integer.
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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