Abstract and Applied Analysis

Volume 2013 (2013), Article ID 871512, 6 pages

http://dx.doi.org/10.1155/2013/871512

Research Article

Umbral Calculus and the Frobenius-Euler Polynomials

Dae San Kim,¹ Taekyun Kim,² and Sang-Hun Lee³

¹Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

²Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

³Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 27 November 2012; Accepted 19 December 2012

Academic Editor: Juan J. Trujillo

Copyright © 2013 Dae San Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study some properties of umbral calculus related to the Appell sequence. From those properties, we derive new and interesting identities of the Frobenius-Euler polynomials.

1. Introduction

Let be the complex number field. For with , the Frobenius-Euler polynomials are defined by the generating function to be

(see [1–5]) with the usual convention about replacing by .

In the special case, are called the th Frobenius-Euler numbers. By (1), we get

(see [6–9]) with the usual convention about replacing by .

Thus, from (1) and (2), we note that

where is the kronecker symbol (see [1, 10, 11]).

For , the Frobenius-Euler polynomials of order are defined by the generating function to be

In the special case, are called the th Frobenius-Euler numbers of order (see [1, 10]).

From (4), we can derive the following equation:

By (5), we see that is a monic polynomial of degree with coefficients in .

Let be the algebra of polynomials in the single variable over and let be the vector space of all linear functionals on . As is known, denotes the action of the linear functional on a polynomial and we remind that the addition and scalar multiplication on are, respectively, defined by

where is a complex constant (see [3, 12]).

Let denote the algebra of formal power series:

(see [3, 12]). The formal power series define a linear functional on by setting

Indeed, by (7) and (8), we get

(see [3, 12]). This kind of algebra is called an umbral algebra.

The order of a nonzero power series is the smallest integer for which the coefficient of does not vanish. A series for which is said to be an invertible series (see [2, 12]). For , and , we have

(see [12]). One should keep in mind that each plays three roles in the umbral calculus: a formal power series, a linear functional, and a linear operator. To illustrate this, let and . As a linear functional, satisfies . As a linear operator, satisfies (see [12]). Let denote a polynomial in with degree . Let us assume that is a delta series and is an invertible series. Then there exists a unique sequence of polynomials such that for all (see [3, 12]). This sequence is called the Sheffer sequence for which is denoted by . If , then is called the associated sequence for . If , then is called the Appell sequence.

Let . Then we see that

where is the compositional inverse of (see [3]). In this paper, we study some properties of umbral calculus related to the Appell sequence. For those properties, we derive new and interesting identities of the Frobenius-Euler polynomials.

2. The Frobenius-Euler Polynomials and Umbral Calculus

By (4) and (12), we see that

Thus, by (13), we get

Let

Then it is an -dimensional vector space over .

So we see that is a basis for . For , let

Then, by (13), (14), and (16), we get

From (17), we have

Therefore, by (16) and (18), we obtain the following theorem.

Theorem 1. For , let

Then one has

where .

From Theorem 1, we note that

Let us consider the operator with and let . Then we have

Thus, by (22), we get

From (4), we can derive

By (23) and (24), we get

From (25), we have

For , from (25), we have

On the other hand, by (12), (13), and (25),

Thus, by (28), we get

Therefore, by (27) and (29), we obtain the following theorem.

Theorem 2. For any , one has

Let us take in Theorem 2. Then we obtain the following corollary.

Corollary 3. For , one has

Let us take in Theorem 2. Then we obtain the following corollary.

Corollary 4. For , one has

Now, we define the analogue of Stirling numbers of the second kind as follows:

Note that is the Stirling number of the second kind.

From the definition of , we have

By (33) and (34), we get

Let us take . Then we have

By (36), we get

Let us take in (37). Then we obtain the following theorem.

Theorem 5. We have

Let us consider in the identity of Theorem 2. Then we have

Let us take in (39). Then we obtain the following theorem.

Theorem 6. For and , one has

Remark 7. Note that

Acknowledgment

The authors would like to express their gratitude to the referees for their valuable suggestions.

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