Research Article | Open Access

# Umbral Calculus and the Frobenius-Euler Polynomials

**Academic Editor:**Juan J. Trujillo

#### 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.

#### References

- T. Kim, “Identities involving Frobenius–Euler polynomials arising from non-linear differential equations,”
*Journal of Number Theory*, vol. 132, no. 12, pp. 2854–2865, 2012. View at: Publisher Site | Google Scholar | MathSciNet - T. Kim and J. Choi, “A note on the product of Frobenius-Euler polynomials arising from the $p$-adic integral on ${Z}_{p}$,”
*Advanced Studies in Contemporary Mathematics*, vol. 22, no. 2, pp. 215–223, 2012. View at: Google Scholar | MathSciNet - S. Roman,
*The Umbral Calculus*, Dover, New York, NY, USA, 2005. View at: MathSciNet - Y. Simsek, O. Yurekli, and V. Kurt, “On interpolation functions of the twisted generalized Frobenius-Euler numbers,”
*Advanced Studies in Contemporary Mathematics*, vol. 15, no. 2, pp. 187–194, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet - K. Shiratani, “On Euler numbers,”
*Memoirs of the Faculty of Science. Kyushu University A*, vol. 27, pp. 1–5, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Araci and M. Acikgoz, “A note on the frobenius-euler numbers and polynomials associated with bernstein polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 22, no. 3, pp. 399–406, 2012. View at: Google Scholar - L. Carlitz, “Some polynomials related to the Bernoulli and Euler polynomials,”
*Utilitas Mathematica*, vol. 19, pp. 81–127, 1981. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. Can, M. Cenkci, V. Kurt, and Y. Simsek, “Twisted Dedekind type sums associated with Barnes' type multiple Frobenius-Euler $l$-functions,”
*Advanced Studies in Contemporary Mathematics*, vol. 18, no. 2, pp. 135–160, 2009. View at: Google Scholar | MathSciNet - I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order $w$-$q$-Genocchi numbers,”
*Advanced Studies in Contemporary Mathematics*, vol. 19, no. 1, pp. 39–57, 2009. View at: Google Scholar | MathSciNet - R. Dere and Y. Simsek, “Applications of umbral algebra to some special polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 22, no. 3, pp. 433–438, 2012. View at: Google Scholar - K. Shiratani and S. Yamamoto, “On a $p$-adic interpolation function for the Euler numbers and its derivatives,”
*Memoirs of the Faculty of Science. Kyushu University A*, vol. 39, no. 1, pp. 113–125, 1985. View at: Publisher Site | Google Scholar | MathSciNet - D. S. Kim and T. Kim, “Some identities of Frobenius-Euler polynomials arising from umbral calculus,”
*Advances in Difference Equations*, vol. 2012, article 196, 2012. View at: Publisher Site | Google Scholar

#### Copyright

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.