AN EXTENSION OF GENERALIZED EULER POLYNOMIALS OF THE SECOND KIND

Abstract

Many mathematicians have studied various relations beween Euler number $E_n$, Bernoulli number $B_n$ and Genocchi number $G_n$ (see [1-18]). They have found numerous important applications in number theory. Howard, T.Agoh, S.-H.Rim have studied Genocchi numbers, Bernoulli numbers, Euler numbers and polynomials of these numbers [1,5,9,15]. T.Kim, M.Cenkci, C.S.Ryoo, L. Jang have studied the q-extension of Euler and Genocchi numbers and polynomials [6,8,10,11,14,17]. In this paper, our aim is introducing and investigating an extension term of generalized Euler polynomials. We also obtain some identities and relations involving the Euler numbers and the Euler polynomials, the Genocchi numbers and Genocchi polynomials.

1. Introduction

The Genocchi number Gn, the Bernoulli number Bn(n ∈ ℕ0 = {0, 1, 2, ... }) and the Euler number En are defined by the following generating function.

For a real or complex parameter α, the generalized Bernoulli polynomials of order α ∈ ℤ, and the generalized Euler polynomials of order α ∈ ℤ are defined by the following generating functions (see, for details, [4, p.253 et seq.], [14, Section 2.8] and [18, Section 1.6]).

and

The Genocchi polynomials Gn(x) of order k ∈ ℕ are defined by

The Euler numbers En and Euler polynomials En(x) are defined by

where [x] is the greatest integer not exceeding x (see [6,8,9,10,11,13,15,16]).

By(1.1), we have

where ℕ is the set of positive integers. The Genocchi number Gn satisfy the recurrence relation

Therefore, we find out that G2 = –1, G4 = 1, G6 = –3, G8 = 17, G10 = –155, G12 = 2073, G14 = –38227, .... That is, G2n+1 = 0(n ≥ 1).

The Stirling number of the first kind s(n, k) can be defined by means of

or by the generating function

We get (1.9) from (1.7) and (1.8)

with s(n, 0) = 0(n > 0), s(n, n) = 1(n ≥ 0), s(n, 1) = (–1)n−1(n–1)!(n > 0), s(n, k) = 0(k > n or k < 0). Stirling number of the second kind S(n, k) can be defined by

or by the generating function

We get (1.12) from (1.10) and (1.11)

with S(n, 0) = (n > 0), S(n, n) = 1(n ≥ 0), S(n, 1) = 1(n > 0), S(n, k) = 0 (k > n or k < 0).

We begin with discussing Euler numbers, Genocchi numbers, Bernoulli numbers, Stirling numbers of the first kind, Stirling numbers of the second kind. In the paper, we organized the entire contents as follows. In Section 2, we define the extension term of generalized Euler polynomials of the second kind and prove them. We also study some interesting relations about a polynomial of x and α with integers coefficients. In Section3, the extension term of generalized Euler polynomials of the second kind will be used to induce the main results of this paper. We also obtain some identities involving the Genocchi numbers, Genocchi polynomials, the Euler numbers, Euler polynomials and prove them.

 

2. Some relations within the an extension terms of the generalized Euler polynomials of the second kind

In this section, we study some relations of the extension term of generalized Euler polynomials of the second kind and research for properties between . First of all, we define the generalized Euler polynomials of the second kind as follows. In [7], we introduced the generalized Euler polynomials of the second kind and investigate their properties. First of all, we introduce the generalized Euler polynomials of the second kind as follows. This completes with the usual convention of replacing (see, for details, [7]).

Definition 2.1. Let x be a real or complex parameter, n ≥ k(n, k ∈ ℕ). Then we define

We derive that

where

For a real and complex parameter α, the generalized Euler polynomials , each of degree n in x as well as in α, are defined by means of the generating function.

Definition 2.2. Let α be a real or complex parameter. Then we define

By using Definition 2.2, we have the addition theorem of polynomials and the relation of polynomials and numbers

Theorem 2.3. (Addition theorem) Let α, x, y ∈ ℂ and n be non-negative inte-gers. Then we get

Proof. For n be non-negative integers, we have

By comparing the coefficients of both side, we complete the proof of the Theorem 2.3.

By using the Definition 2.2, we have the following Theorem 2.4.

Theorem 2.4. Let n ≥ k(n, k, l ∈ ℕ). Then we derive that

where

Proof. By (2.1) and (2.2), we easily have

By Definition 2.2, (1.4) and (1.8) we have

which readily yields

Therefore, we complete the proof of Theorem 2.4

Remark 2.1. From (2.3) and Theorem 2.4, we find out that

ρ(α)(0, 0) = 1,

ρ(α)(1, 0) = 0, ρ(α)(1, 1) = 1,

ρ(α)(2, 0) = –α, ρ(α)(2, 1) = 0, ρ(α)(2, 2) = 1,

ρ(α)(3, 0) = 0, ρ(α)(3, 1) = –3α, ρ(α)(3, 2) = 0, ρ(α)(3, 3) = 1,

ρ(α)(4, 0) = 2α+3α2, ρ(α)(4, 1) = 0, ρ(α)(4, 2) = –6α, ρ(α)(4, 3) = 0, ρ(α)(4, 4) = 1, . . .

Thus, we know that is a polynomial of x. Setting n = 1, 2, 3, 4, 5 in Theorem 2.4, we get to

We also find out an extension terms of the generalized Euler polynomials that can be represented by c(n, k) with Stirling numbers of the first kind, Stirling numbers of the second kind.

Since

we have the following theorem.

Theorem 2.5. Let n, k ∈ ℕ, then by (1.1) and Definition 2.2, we have

 

3. Some relations between an extension of the generalized Euler polynomials and Euler numbers, Genocchi numbers and themselves

In this section, we access some relations between an extension terms of generalized Euler polynomials and Euler numbers, Euler polynomials, Genocchi numbers, Genocchi polynomials of order k. We construct relations among an extension terms of generalized Euler polynomials themselves.

Theorem 3.1. Let n ≥ k(n, k ∈ ℕ). Relation between and Genocchi numbers Gn, we have

where

Proof. By (1.1), we have

Let us that

Therefore, we have

And we expand to degree of α, we obtain

We deduce that by the generalized Euler polynomials

By (2.4), we may immediately obtain Theorem 3.1. This completes the proof of Theorem 3.1.

We find out that G∗(0) = 1, G∗(1) = 0, G∗(2) = –1, G∗(3) = 0, G∗(4) = 5, G∗(5) = 6, G∗(6) = –61, ....

Remark 3.1. Let n ≥ k(n, k ∈ ℕ). Then we have

where

Theorem 3.2. Let n ≥ k(n, k ∈ ℕ). Then we obtain

Proof. By applying Theorem 2.4, we have

It follows from Definition 2.2 that

On the other hand, we have from (2.1)

Substituting (3.9) in (3.8) we get

By (3.10), we may immediately obtain Theorem 3.2. This completes the proof of Theorem 3.2.

Theorem 3.3. Let n ≥ k(n, k ∈ ℕ). Relation between and Euler num-bers En, we have

where

Proof. By definition (1.1)

Therefore, according to (3.1), (3.2), we have

By Theorem 3.1, we may immediately obtain Theorem 3.3.

We find out that E∗(0) = 1, E∗(1) = 0, E∗(2) = –1, E∗(3) = 0, E∗(4) = 5, E∗(5) = 0, E∗(6) = –61, .... Thus, we easily see that G∗= E∗.

By Definition 2.2, we have

Therefore, we have the following theorem.

Theorem 3.4. Let α be a real or complex parameter. Relation between and Euler polynomials , we have

For α = 1 in (3.13), we have the following corollary.

Corollary 3.5. For α = 1, we have

Theorem 3.6. Let n, α ∈ ℕ. Relation between and Genocchi polynomi-als we have

Proof. By Definition 2.2,

Therefore, we may immediately obtain Theorem 3.6. This completes the proof of Theorem 3.6.