1.Introduction DaeSanKim, TaekyunKim, andSang-HunLee UmbralCalculusandtheFrobenius-EulerPolynomials ResearchArticle

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Volume 2013, Article ID 871512,6pages 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 Lee

³

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

Correspondence should be addressed to Taekyun Kim; [email protected] 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.

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

LetC be the complex number field. For𝜆 ∈C with𝜆 ̸= 1, the Frobenius-Euler polynomials are defined by the generating function to be

1 − 𝜆

𝑒^𝑡− 𝜆𝑒^𝑥𝑡= 𝑒^{𝐻(𝑥|𝜆)𝑡}=∑^∞

𝑛=0

𝐻_𝑛(𝑥 | 𝜆)𝑡^𝑛

𝑛!, (1)

(see [1–5]) with the usual convention about replacing𝐻^𝑛(𝑥 | 𝜆)by𝐻_𝑛(𝑥 | 𝜆).

In the special case,𝑥 = 0, 𝐻_𝑛(0 | 𝜆) = 𝐻_𝑛(𝜆)are called the𝑛th Frobenius-Euler numbers. By (1), we get

𝐻_𝑛(𝑥 | 𝜆) =∑^𝑛

𝑙=0(𝑛𝑙)𝐻^𝑛−𝑙(𝜆) 𝑥^𝑙= (𝐻 (𝜆) + 𝑥)^𝑛, (2) (see [6–9]) with the usual convention about replacing𝐻^𝑛(𝜆) by𝐻_𝑛(𝜆).

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

(𝐻 (𝜆) + 1)^𝑛− 𝜆𝐻_𝑛(𝜆) = (1 − 𝜆) 𝛿_0,𝑛, (3)

where𝛿_𝑛,𝑘is the kronecker symbol (see [1,10,11]).

For𝑟 ∈ Z₊, the Frobenius-Euler polynomials of order𝑟 are defined by the generating function to be

(1 − 𝜆

𝑒^𝑡− 𝜆)^𝑟𝑒^𝑥𝑡= (1 − 𝜆

𝑒^𝑡− 𝜆) × ⋅ ⋅ ⋅ × (1 − 𝜆 𝑒^𝑡− 𝜆)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟𝑒^𝑥𝑡

𝑟-times

=∑^∞

𝑛=0𝐻_𝑛^(𝑟)(𝑥 | 𝜆)𝑡^𝑛 𝑛!.

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In the special case,𝑥 = 0, 𝐻_𝑛^(𝑟)(0 | 𝜆) = 𝐻_𝑛^(𝑟)(𝜆)are called the 𝑛th Frobenius-Euler numbers of order𝑟(see [1,10]).

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

𝐻_𝑛^(𝑟)(𝑥 | 𝜆) =∑^𝑛

𝑙=0(𝑛𝑙)𝐻^𝑛−𝑙^(𝑟)(𝜆) 𝑥^𝑙, 𝐻_𝑛^(𝑟)(𝜆) = ∑

𝑙₁+⋅⋅⋅+𝑙_𝑟=𝑛

( 𝑛

𝑙₁, . . . , 𝑙_𝑟) 𝐻_𝑙₁(𝜆) ⋅ ⋅ ⋅ 𝐻_𝑙_𝑟(𝜆) . (5)

By (5), we see that𝐻^(𝑟)_𝑛 (𝑥 | 𝜆)is a monic polynomial of degree 𝑛with coefficients inQ(𝜆).

LetPbe the algebra of polynomials in the single variable 𝑥overC and letP^∗be the vector space of all linear functionals onP. As is known, ⟨𝐿 | 𝑝(𝑥)⟩denotes the action of the linear functional𝐿on a polynomial𝑝(𝑥)and we remind that

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the addition and scalar multiplication onP^∗are, respectively, defined by

⟨𝐿 + 𝑀 | 𝑝 (𝑥)⟩ = ⟨𝐿 | 𝑝 (𝑥)⟩ + ⟨𝑀 | 𝑝 (𝑥)⟩,

⟨𝑐𝐿 | 𝑝 (𝑥)⟩ = 𝑐⟨𝐿 | 𝑝 (𝑥)⟩, (6) where𝑐is a complex constant (see [3,12]).

LetF denote the algebra of formal power series:

F= {𝑓 (𝑡) =∑^∞

𝑘=0

𝑎_𝑘

𝑘!𝑡^𝑘| 𝑎_𝑘∈C} (7) (see [3,12]). The formal power series define a linear functional onPby setting

⟨𝑓 (𝑡) | 𝑥^𝑛⟩ = 𝑎_𝑛, ∀𝑛 ≥ 0. (8) Indeed, by (7) and (8), we get

⟨𝑡^𝑘| 𝑥^𝑛⟩ = 𝑛!𝛿_𝑛,𝑘 (𝑛, 𝑘 ≥ 0) (9) (see [3,12]). This kind of algebra is called an umbral algebra.

The order𝑂(𝑓(𝑡))of a nonzero power series𝑓(𝑡)is the smallest integer𝑘 for which the coefficient of 𝑡^𝑘 does not vanish. A series𝑓(𝑡)for which𝑂(𝑓(𝑡)) = 1is said to be an invertible series (see [2,12]). For𝑓(𝑡), 𝑔(𝑡) ∈F, and𝑝(𝑥) ∈P, we have

⟨𝑓 (𝑡) 𝑔 (𝑡) | 𝑝 (𝑥)⟩ = ⟨ 𝑓 (𝑡) | 𝑔 (𝑡) 𝑝 (𝑥)⟩

= ⟨ 𝑔 (𝑡) | 𝑓 (𝑡) 𝑝 (𝑥)⟩ (10) (see [12]). One should keep in mind that each𝑓(𝑡) ∈F plays three roles in the umbral calculus: a formal power series, a linear functional, and a linear operator. To illustrate this, let 𝑝(𝑥) ∈ Pand𝑓(𝑡) = 𝑒^𝑦𝑡 ∈ F. As a linear functional,𝑒^𝑦𝑡 satisfies⟨𝑒^𝑦𝑡| 𝑝(𝑥)⟩ = 𝑝(𝑦). As a linear operator,𝑒^𝑦𝑡satisfies 𝑒^𝑦𝑡𝑝(𝑥) = 𝑝(𝑥 + 𝑦)(see [12]). Let𝑠_𝑛(𝑥)denote a polynomial in𝑥with degree𝑛. Let us assume that𝑓(𝑡)is a delta series and 𝑔(𝑡)is an invertible series. Then there exists a unique sequence𝑠_𝑛(𝑥)of polynomials such that⟨𝑔(𝑡)𝑓(𝑡)^𝑘| 𝑠_𝑛(𝑥)⟩ = 𝑛!𝛿_𝑛,𝑘 for all 𝑛, 𝑘 ≥ 0(see [3, 12]). This sequence 𝑠_𝑛(𝑥) is called the Sheffer sequence for(𝑔(𝑡), 𝑓(𝑡))which is denoted by𝑠_𝑛(𝑥) ∼ (𝑔(𝑡), 𝑓(𝑡)). If𝑠_𝑛(𝑥) ∼ (1, 𝑓(𝑡)), then𝑠_𝑛(𝑥)is called the associated sequence for𝑓(𝑡). If 𝑠_𝑛(𝑥) ∼ (𝑔(𝑡), 𝑡), then 𝑠_𝑛(𝑥)is called the Appell sequence.

Let𝑠_𝑛(𝑥) ∼ (𝑔(𝑡), 𝑓(𝑡)). Then we see that ℎ (𝑡) =∑^∞

𝑘=0

⟨ℎ (𝑡) | 𝑠_𝑘(𝑥)⟩

𝑘! 𝑔 (𝑡) 𝑓(𝑡)^𝑘, ℎ (𝑡) ∈F, 𝑝 (𝑥) =∑^∞

𝑘=0

⟨𝑔 (𝑡) 𝑓(𝑡)^𝑘| 𝑝 (𝑥)⟩

𝑘! 𝑠_𝑘(𝑥) , 𝑝 (𝑥) ∈P, 𝑓 (𝑡) 𝑠_𝑛(𝑥) = 𝑛𝑠_𝑛−1(𝑥) ,

⟨𝑓 (𝑡) | 𝑝 (𝛼𝑥)⟩ = ⟨ 𝑓 (𝛼𝑡 | 𝑝 (𝑥)⟩ ,

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1

𝑔 (𝑓 (𝑡))𝑒^{𝑦𝑓(𝑡)}=∑^∞

𝑘=0

𝑠_𝑘(𝑦)

𝑘! 𝑡^𝑘, ∀𝑦 ∈C, (12)

where 𝑓(𝑡) is the compositional inverse of 𝑓(𝑡) (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

𝐻_𝑛^(𝑟)(𝑥 | 𝜆) ∼ ((𝑒^𝑡− 𝜆

1 − 𝜆)^𝑟, 𝑡) . (13) Thus, by (13), we get

⟨(𝑒^𝑡− 𝜆

1 − 𝜆)^𝑟𝑡^𝑘| 𝐻_𝑛^(𝑟)(𝑥 | 𝜆)⟩ = 𝑛!𝛿_𝑛,𝑘. (14) Let

P_𝑛(𝜆) = {𝑝 (𝑥) ∈Q(𝜆) [𝑥] |deg𝑝 (𝑥) ≤ 𝑛} . (15) Then it is an(𝑛 + 1)-dimensional vector space overQ(𝜆).

So we see that{𝐻₀^(𝑟)(𝑥 | 𝜆), 𝐻₁^(𝑟)(𝑥 | 𝜆), . . . , 𝐻_𝑛^(𝑟)(𝑥 | 𝜆)}is a basis forP_𝑛(𝜆). For𝑝(𝑥) ∈P_𝑛(𝜆), let

𝑝 (𝑥) =∑^𝑛

𝑘=0

𝐶_𝑘𝐻_𝑘^(𝑟)(𝑥 | 𝜆) , (𝑛 ≥ 0) . (16) Then, by (13), (14), and (16), we get

⟨(𝑒^𝑡− 𝜆

1 − 𝜆)^𝑟𝑡^𝑘| 𝑝 (𝑥)⟩

=∑^𝑛

𝑙=0

𝐶_𝑙⟨(𝑒^𝑡− 𝜆 1 − 𝜆)

𝑟

𝑡^𝑘| 𝐻_𝑙^(𝑟)(𝑥 | 𝜆)⟩

=∑^𝑛

𝑙=0

𝐶_𝑙𝑙!𝛿_𝑙,𝑘= 𝑘!𝐶_𝑘.

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From (17), we have 𝐶_𝑘= 1

𝑘!⟨(𝑒^𝑡− 𝜆 1 − 𝜆)

𝑟

𝑡^𝑘| 𝑝 (𝑥)⟩

= 1

𝑘!⟨(𝑒^𝑡− 𝜆

1 − 𝜆)^𝑟 | 𝐷^𝑘𝑝 (𝑥)⟩

= 1

𝑘!(1 − 𝜆)^𝑟

∑𝑟

𝑗=0(𝑟𝑗)(−𝜆)^𝑟−𝑗⟨𝑒^𝑗𝑡| 𝐷^𝑘𝑝 (𝑥)⟩

= 1

𝑘!(1 − 𝜆)^𝑟

∑𝑟

𝑗=0(𝑟𝑗)(−𝜆)^𝑟−𝑗⟨𝑡⁰| 𝑒^𝑗𝑡𝐷^𝑘𝑝 (𝑥)⟩

= 1

𝑘!(1 − 𝜆)^𝑟

∑𝑟

𝑗=0(𝑟𝑗)(−𝜆)^𝑟−𝑗⟨𝑡⁰| 𝐷^𝑘𝑝 (𝑥 + 𝑗)⟩ . (18)

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

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Theorem 1. For𝑝(𝑥) ∈P_𝑛(𝜆), let 𝑝 (𝑥) =∑^𝑛

𝑘=0

𝐶_𝑘𝐻_𝑘^(𝑟)(𝑥) . (19) Then one has

𝐶_𝑘= 1 𝑘!(1 − 𝜆)^𝑟

∑𝑟

𝑗=0(𝑟𝑗)(−𝜆)^𝑟−𝑗𝐷^𝑘𝑝 (𝑗) , (20) where𝐷𝑝(𝑥) = 𝑑𝑝(𝑥)/𝑑𝑥.

FromTheorem 1, we note that 𝑝 (𝑥) = 1

(1 − 𝜆)^𝑟

⋅∑^𝑛

𝑘=0

{{ {

∑𝑟 𝑗=0

1

𝑘!(𝑟𝑗)(−𝜆)^𝑟−𝑗𝐷^𝑘𝑝 (𝑗)} }}

𝐻_𝑘^(𝑟)(𝑥 | 𝜆) . (21) Let us consider the operator ̃Δ_𝜆with̃Δ_𝜆𝑓(𝑥) = 𝑓(𝑥 + 1) − 𝜆𝑓(𝑥)and let𝐽_𝜆= (1/(1 − 𝜆))̃Δ_𝜆. Then we have

𝐽_𝜆(𝑓) (𝑥) = 1

1 − 𝜆{𝑓 (𝑥 + 1) − 𝜆𝑓 (𝑥)} . (22) Thus, by (22), we get

𝐽_𝜆(𝐻_𝑛^(𝑟)(𝑥 | 𝜆)) = 1

1 − 𝜆{𝐻_𝑛^(𝑟)(𝑥 + 1 | 𝜆) − 𝜆𝐻_𝑛^(𝑟)(𝑥 | 𝜆)} . (23) From (4), we can derive

∑∞ 𝑛=0

{𝐻_𝑛^(𝑟)(𝑥 + 1 | 𝜆) − 𝜆𝐻_𝑛^(𝑟)(𝑥 | 𝜆)}𝑡^𝑛 𝑛!

= (1 − 𝜆

𝑒^𝑡− 𝜆)^𝑟𝑒^(𝑥+1)𝑡− 𝜆(1 − 𝜆 𝑒^𝑡− 𝜆)^𝑟𝑒^𝑥𝑡

= (1 − 𝜆

𝑒^𝑡− 𝜆)^𝑟𝑒^𝑥𝑡(𝑒^𝑡− 𝜆) = (1 − 𝜆) (1 − 𝜆 𝑒^𝑡− 𝜆)^𝑟−1𝑒^𝑥𝑡

= (1 − 𝜆)∑^∞

𝑛=0𝐻_𝑛^(𝑟−1)(𝑥 | 𝜆)𝑡^𝑛 𝑛!.

(24) By (23) and (24), we get

𝐽_𝜆(𝐻_𝑛^(𝑟)(𝑥 | 𝜆)) = 𝐻_𝑛^(𝑟−1)(𝑥 | 𝜆) . (25) From (25), we have

𝐽_𝜆^𝑟(𝐻_𝑛^(𝑟)(𝑥 | 𝜆)) = 𝐽_𝜆^𝑟−1(𝐻_𝑛^(𝑟−1)(𝑥 | 𝜆))

= ⋅ ⋅ ⋅ = 𝐻_𝑛⁽⁰⁾(𝑥 | 𝜆) = 𝑥^𝑛, 𝐽_𝜆^𝑟(𝑥^𝑛) = 𝐽_𝜆^𝑟𝐻_𝑛⁽⁰⁾(𝑥 | 𝜆) = 𝐻_𝑛^(−𝑟)(𝑥 | 𝜆) = 𝐽_𝜆^2𝑟𝐻_𝑛^(𝑟)(𝑥 | 𝜆) .

(26)

For𝑠 ∈Z₊, from (25), we have

𝐽_𝜆^𝑠(𝐻_𝑛^(𝑟)(𝑥 | 𝜆)) = 𝐻_𝑛^{(𝑟−𝑠)}(𝑥 | 𝜆) . (27) On the other hand, by (12), (13), and (25),

𝐽_𝜆^𝑠(𝐻_𝑛^(𝑟)(𝑥 | 𝜆)) = (𝑒^𝑡− 𝜆 1 − 𝜆)

𝑠

(𝐻_𝑛^(𝑟)(𝑥 | 𝜆))

= 1

(1 − 𝜆)^𝑠((1 − 𝜆) +∑^∞

𝑘=1

𝑡^𝑘 𝑘!)

𝑠

⋅ (𝐻_𝑛^(𝑟)(𝑥 | 𝜆)) .

(28)

Thus, by (28), we get

𝐽_𝜆^𝑠(𝐻_𝑛^(𝑟)(𝑥 | 𝜆))

= ∑^𝑠

𝑚=0

(_𝑚^𝑠) (1 − 𝜆)^𝑚

∑∞ 𝑙=𝑚

( ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘𝑗≥1

1

𝑘₁! ⋅ ⋅ ⋅ 𝑘_𝑚!) 𝑡^𝑙(𝐻_𝑛^(𝑟)(𝑥 | 𝜆))

= ∑^𝑠

𝑚=0

(_𝑚^𝑠) (1 − 𝜆)^𝑚

∑∞ 𝑙=𝑚

1

𝑙!( ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) 𝐷^𝑙)

⋅ 𝐻_𝑛^(𝑟)(𝑥 | 𝜆)

=^{min{𝑠,𝑛}}∑

𝑚=0

(_𝑚^𝑠) (1 − 𝜆)^𝑚

∑𝑛

𝑙=𝑚(𝑛𝑙) ∑_𝑘

1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘_𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) 𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆)

=^{min{𝑠,𝑛}}∑

𝑙=0

{{ {{ {{ {

(𝑛𝑙)∑^𝑙

𝑚=0

(_𝑚^𝑠) (1 − 𝜆)^𝑚

⋅ ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆)

+ ∑^𝑛

𝑙=min{𝑠,𝑛}+1

{{ {{ {{ {

(𝑛𝑙)^{min{𝑠,𝑛}}∑

𝑚=0

(_𝑚^𝑠) (1 − 𝜆)^𝑚

⋅ ∑

𝑘₁+⋅⋅⋅+𝑘_𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆) .

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

(4)

Theorem 2. For any𝑟, 𝑠 ≥ 0, one has 𝐻_𝑛^{(𝑟−𝑠)}(𝑥 | 𝜆)

=^{min{𝑠,𝑛}}∑

𝑙=0

{{ {{ {{ {

(𝑛𝑙)∑^𝑙

𝑚=0

(_𝑚^𝑠) (1 − 𝜆)^𝑚 ∑

𝑘₁+⋅⋅⋅+𝑘_𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

⋅ 𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆)

+ ∑^𝑛

𝑙=min{𝑠,𝑛}+1

{{ {{ {{ {

(𝑛𝑙)^{min{𝑠,𝑛}}∑

𝑚=0

(_𝑚^𝑠) (1 − 𝜆)^𝑚

⋅ ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘_𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆) .

(30) Let us take𝑠 = 𝑟−1 (𝑟 ≥ 1)inTheorem 2. Then we obtain the following corollary.

Corollary 3. For𝑛 ≥ 0, 𝑟 ≥ 1, one has 𝐻_𝑛(𝑥 | 𝜆)

=^{min{𝑟−1,𝑛}}∑

𝑙=0

{{ {{ {{ {

(𝑛𝑙)∑^𝑙

𝑚=0

(^𝑟−1_𝑚 ) (1 − 𝜆)^𝑚 ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

⋅ 𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆)

+ ∑^𝑛

𝑙=min{𝑟−1,𝑛}+1

{{ {{ {{ {

(𝑛𝑙)^{min{𝑟−1,𝑛}}∑

𝑚=0

(^𝑟−1_𝑚 ) (1 − 𝜆)^𝑚

⋅ ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆) .

(31) Let us take𝑠 = 𝑟 (𝑟 ≥ 1)inTheorem 2. Then we obtain the following corollary.

Corollary 4. For𝑛 ≥ 0, 𝑟 ≥ 1, one has

𝑥^𝑛 =^{min{𝑟,𝑛}}∑

𝑙=0

{{ {{ {{ {

(𝑛𝑙)∑^𝑙

𝑚=0

(_𝑚^𝑟) (1 − 𝜆)^𝑚 ∑

𝑘₁+⋅⋅⋅+𝑘_𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

⋅ 𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆)

+ ∑^𝑛

𝑙=min{𝑟,𝑛}+1

{{ {{ {{ {

(𝑛𝑙)^{min{𝑟,𝑛}}∑

𝑚=0

(_𝑚^𝑟) (1 − 𝜆)^𝑚

⋅ ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆) .

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

𝑆_𝜆(𝑛, 𝑘) = 1 𝑘!

∑𝑘

𝑗=0(𝑘𝑗)(−𝜆)^𝑘−𝑗𝑗^𝑛, (𝑛, 𝑘 ≥ 0) . (33) Note that 𝑆₁(𝑛, 𝑘) = 𝑆(𝑛, 𝑘) is the Stirling number of the second kind.

From the definition of̃Δ_𝜆, we have

̃Δ^𝑛_𝜆𝑓 (0) =∑^𝑛

𝑘=0(𝑛𝑘)(−𝜆)^𝑛−𝑘𝑓 (𝑘) . (34) By (33) and (34), we get

𝑆_𝜆(𝑛, 𝑘) = 1

𝑘!̃Δ^𝑘_𝜆0^𝑛, (𝑛, 𝑘 ≥ 0) . (35) Let us take𝑠 = 2𝑟. Then we have

𝐽_𝜆^𝑟𝑥^𝑛

= 𝐻_𝑛^(−𝑟)(𝑥 | 𝜆)

=^{min{2𝑟,𝑛}}∑

𝑙=0

{{ {{ {{ {

(𝑛𝑙)∑^𝑙

𝑚=0

(^2𝑟_𝑚) (1 − 𝜆)^𝑚 ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘_𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

⋅ 𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆)

+ ∑^𝑛

𝑙=min{2𝑟,𝑛}+1

{{ {{ {{ {

(𝑛𝑙)^{min{2𝑟,𝑛}}∑

𝑚=0

(^2𝑟_𝑚) (1 − 𝜆)^𝑚

⋅ ∑

𝑘₁+⋅⋅⋅+𝑘_𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆) ,

𝐽_𝜆^𝑟𝑥^𝑛= ( 1

1 − 𝜆̃Δ_𝜆)^𝑟𝑥^𝑛

= 1

(1 − 𝜆)^𝑟

∑𝑟

𝑗=0(𝑟𝑗)(−𝜆)^𝑟−𝑗(𝑥 + 𝑗)^𝑛.

(36)

(5)

By (36), we get 1 (1 − 𝜆)^𝑟

∑𝑟

𝑗=0(𝑟𝑗)(−𝜆)^𝑟−𝑗(𝑥 + 𝑗)^𝑛

= 1

(1 − 𝜆)^𝑟̃Δ^𝑟_𝜆𝑥^𝑛

=^{min{2𝑟,𝑛}}∑

𝑙=0

{{ {{ {{ {

(𝑛𝑙)∑^𝑙

𝑚=0

(^2𝑟_𝑚) (1 − 𝜆)^𝑚 ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

⋅ 𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆)

+ ∑^𝑛

{{ {{ {{ {

𝑚=0

(^2𝑟_𝑚) (1 − 𝜆)^𝑚

⋅ ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }} }

𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆) .

(37) Let us take 𝑥 = 0in (37). Then we obtain the following theorem.

Theorem 5. We have 𝑟!

(1 − 𝜆)^𝑟𝑆_𝜆(𝑛, 𝑟)

= 𝑟!

(1 − 𝜆)^𝑟

̃Δ^𝑟_𝜆0^𝑛 𝑟!

=^{min{2𝑟,𝑛}}∑

𝑙=0

{{ {{ {{ {{ {{ {

(𝑛𝑙)∑^𝑙

𝑚=0

(^2𝑟_𝑚) (1 − 𝜆)^𝑚 ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }} }} }

⋅ 𝐻_𝑛−𝑙^(𝑟)(𝜆)

+ ∑^𝑛

{{ {{ {{ {

𝑚=0

(^2𝑟_𝑚) (1 − 𝜆)^𝑚

⋅ ∑

𝑘₁+⋅⋅⋅+𝑘_𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

𝐻_𝑛−𝑙^(𝑟)(𝜆)

=^{min{𝑟,𝑛}}∑

𝑚=0

(_𝑚^𝑟) (1 − 𝜆)^𝑚 ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑛 𝑘_𝑗≥1

( 𝑛

𝑘₁, . . . , 𝑘_𝑚) .

(38)

Let us consider𝑠 = 2𝑟 − 1in the identity ofTheorem 2.

Then we have 𝐽_𝜆^𝑟−1𝑥^𝑛

= 𝐻_𝑛^{−(𝑟−1)}(𝑥 | 𝜆)

=min{2𝑟−1,𝑛}

∑

𝑙=0

{{ {{ {{ {

(𝑛𝑙)∑^𝑙

𝑚=0

(^2𝑟−1_𝑚 ) (1 − 𝜆)^𝑚 ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

⋅ 𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆)

+ ∑^𝑛

𝑙=min{2𝑟−1,𝑛}+1

{{ {{ {{ {

(𝑛𝑙)min{2𝑟−1,𝑛}

∑

𝑚=0

(^2𝑟−1_𝑚 ) (1 − 𝜆)^𝑚

⋅ ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

𝐻_𝑛−𝑙^(𝑟)(𝑥 | 𝜆)

= 1

(1 − 𝜆)^𝑟−1

𝑟−1∑

𝑗=0(𝑟 − 1𝑗 ) (−𝜆)^{𝑟−1−𝑗}(𝑥 + 𝑗)^𝑛

= 1

(1 − 𝜆)^𝑟−1̃Δ^𝑟−1_𝜆 𝑥^𝑛.

(39) Let us take 𝑥 = 0in (39). Then we obtain the following theorem.

Theorem 6. For𝑛 ≥ 0and𝑟 ≥ 1, one has (𝑟 − 1)!

(1 − 𝜆)^𝑟−1𝑆_𝜆(𝑛, 𝑟 − 1)

= (𝑟 − 1)!

(1 − 𝜆)^𝑟−1

̃Δ^𝑟−1_𝜆 0^𝑛 (𝑟 − 1)!

=min{2𝑟−1,𝑛}

∑

𝑙=0

{{ {{ {{ {

(𝑛𝑙)∑^𝑙

𝑚=0

(^2𝑟−1_𝑚 ) (1 − 𝜆)^𝑚 ∑

𝑘₁+⋅⋅⋅+𝑘_𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

⋅ 𝐻_𝑛−𝑙^(𝑟)(𝜆)

+ ∑^𝑛

𝑙=min{2𝑟−1,𝑛}+1

{{ {{ {{ {

(𝑛𝑙)min{2𝑟−1,𝑛}

∑

𝑚=0

(^2𝑟−1_𝑚 ) (1 − 𝜆)^𝑚

⋅ ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘_𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

𝐻_𝑛−𝑙^(𝑟)(𝜆) .

(40)

(6)

Remark 7. Note that (𝑟 − 1)!

(1 − 𝜆)^𝑟−1𝑆_𝜆(𝑛, 𝑟 − 1)

=^{min{𝑟,𝑛}}∑

𝑙=0

{{ {{ {{ {

(𝑛𝑙)∑^𝑙

𝑚=0

(_𝑚^𝑟) (1 − 𝜆)^𝑚 ∑

𝑘1+⋅⋅⋅+𝑘𝑚=𝑙 𝑘_𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

⋅ 𝐻_𝑛−𝑙(𝜆)

+ ∑^𝑛

𝑙=min{𝑟,𝑛}+1

{{ {{ {{ {

(𝑛𝑙)^{min{𝑟,𝑛}}∑

𝑚=0

(_𝑚^𝑟) (1 − 𝜆)^𝑚

⋅ ∑

𝑘₁+⋅⋅⋅+𝑘_𝑚=𝑙 𝑘𝑗≥1

( 𝑙

𝑘₁, . . . , 𝑘_𝑚) }} }} }} }

𝐻_𝑛−𝑙(𝜆) .

(41)

Acknowledgment

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

References

[1] 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.

[2] T. Kim and J. Choi, “A note on the product of Frobenius-Euler polynomials arising from the𝑝-adic integral on𝑍_𝑝,”Advanced Studies in Contemporary Mathematics, vol. 22, no. 2, pp. 215–

223, 2012.

[3] S. Roman,The Umbral Calculus, Dover, New York, NY, USA, 2005.

[4] 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.

[5] K. Shiratani, “On Euler numbers,”Memoirs of the Faculty of Science. Kyushu University A, vol. 27, pp. 1–5, 1973.

[6] 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.

[7] L. Carlitz, “Some polynomials related to the Bernoulli and Euler polynomials,”Utilitas Mathematica, vol. 19, pp. 81–127, 1981.

[8] M. Can, M. Cenkci, V. Kurt, and Y. Simsek, “Twisted Dedekind type sums associated with Barnes’ type multiple Frobenius- Euler𝑙-functions,”Advanced Studies in Contemporary Mathe- matics, vol. 18, no. 2, pp. 135–160, 2009.

[9] 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.

[10] 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.

[11] 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.

[12] 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.

(7)

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1.Introduction DaeSanKim, TaekyunKim, andSang-HunLee UmbralCalculusandtheFrobenius-EulerPolynomials ResearchArticle

Research Article

Umbral Calculus and the Frobenius-Euler Polynomials

Dae San Kim,

Taekyun Kim,

and Sang-Hun Lee

1. Introduction

2. The Frobenius-Euler Polynomials and Umbral Calculus

Acknowledgment

References

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Mathematics

Complex Analysis

Optimization

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