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Frobenius-Euler polynomials and umbral calculus in the p-adic case
Advances in Difference Equations volume 2012, Article number: 222 (2012)
Abstract
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.
MSC:05A10, 05A19.
1 Introduction
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.
For with , let
Note that the natural map induces
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.
For and with , the Frobenius-Euler measure on X is defined by
where the p-adic absolute value on is normalized by .
As is well known, the Frobenius-Euler polynomials are defined by the generating function to be
with the usual convention about replacing by . In the special case, , are called the n th Frobenius-Euler numbers
Thus, by (1.2) and (1.3), we easily get
where is the Kronecker symbol.
For , the Frobenius-Euler polynomials of order r are defined by the generating function
In the special case, , are called the n th Frobenius-Euler numbers of order r. The n th Frobenius-Euler polynomials can be represented by (1.1) as follows:
Let ℱ be the set of all formal power series in the variable t over with
Let and denote the vector space of all linear functionals on ℙ.
The formal power series
defines a linear functional on ℙ by setting
From (1.8) and (1.9), we have
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]).
The order of power series (≠0) is the smallest integer k for which does not vanish (see [11, 15]). A series for which is called a delta series. If a series has , then is called an invertible series (see [11, 15]). Let . Then we easily see that . From (1.10), we note that
and
For , we have
By (1.13), we get
and
Thus, by (1.15), we get
By (1.16), we easily see that
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 [15]).
For , we have
If , then we have
and
where is compositional inverse of (see [11, 15]). In [9], 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
Let
Then we see that is an invertible series. From (1.2), we have
Hence, by (2.2), we get
By (2.2) and (2.3), we get
From (1.6), we have
and
By (2.5), we get
From (1.6) and (2.6), we have
From (2.2), we can easily derive
By (2.8), we get
Thus, from (2.9), we have
By (2.10), we get
From (2.11), we note that
Let us consider the linear functional such that
for all polynomials can be determined from (1.12) to be
By (2.4) and (2.14), we get
Therefore, by (2.15), we obtain the following theorem.
Theorem 2.1 For , we have
In particular,
From (1.6), we have
By (1.6), (2.4) and (2.16), we get
Therefore, by (2.17), we obtain the following theorem.
Theorem 2.2 For , we have
In particular,
By (1.6) and (2.16), we get
From Appell identity and (2.18), we can derive the following identities:
Let
Then is an invertible functional in ℱ. By (1.5) and (2.20), we get
Thus, from (2.21), we have
and
By (2.22) and (2.23), we see that
From (2.4), we can derive the following identity:
By (1.10) and (2.25), we get
From (1.14), we have
By (2.26) and (2.27), we get
where . From (2.25), we note that
Thus, by (2.28), we get
By (2.29), we see that
Therefore, by (2.30), we obtain the following theorem.
Theorem 2.3 For and , we have
In particular,
Moreover,
Let us consider the function in ℱ such that
for all polynomials can be determined from (1.12) to be
Therefore, by (2.31) and (2.32), we obtain the following theorem.
Theorem 2.4 For , we have
In particular,
Indeed, the n th Frobenius-Euler number of order r is given by
where .
Remark From (1.2) and (1.5), we note that
and
Continuing this process, we obtain the following equation:
By (1.2), (1.5) and (2.35), we get
where k is a positive integer.
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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., Lee, SH. et al. Frobenius-Euler polynomials and umbral calculus in the p-adic case. Adv Differ Equ 2012, 222 (2012). https://doi.org/10.1186/1687-1847-2012-222
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DOI: https://doi.org/10.1186/1687-1847-2012-222