#### 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.