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AN EXTENSION OF GENERALIZED EULER POLYNOMIALS OF THE SECOND KIND
AN EXTENSION OF GENERALIZED EULER POLYNOMIALS OF THE SECOND KIND
Journal of Applied Mathematics & Informatics. 2014. Sep, 32(3_4): 465-474
Copyright © 2014, Korean Society of Computational and Applied Mathematics
  • Received : October 30, 2013
  • Accepted : January 20, 2014
  • Published : September 28, 2014
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About the Authors
Y. H. KIM
H. Y. JUNG
C. S. RYOO

Abstract
Many mathematicians have studied various relations beween Euler number En , Bernoulli number Bn and Genocchi number Gn (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 - 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. AMS Mathematics Subject Classification : 11B68, 11S40, 11S80.
Keywords
1. Introduction
The Genocchi number Gn , the Bernoulli number Bn ( n ∈ ℕ 0 = {0, 1, 2, ... }) and the Euler number En are defined by the following generating function.
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For a real or complex parameter α , the generalized Bernoulli polynomials
PPT Slide
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of order α ∈ ℤ, and the generalized Euler polynomials
PPT Slide
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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]).
PPT Slide
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and
PPT Slide
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The Genocchi polynomials Gn ( x ) of order k ∈ ℕ are defined by
PPT Slide
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The Euler numbers En and Euler polynomials En ( x ) are defined by
PPT Slide
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where [ x ] is the greatest integer not exceeding x (see [6 , 8 , 9 , 10 , 11 , 13 , 15 , 16] ).
By(1.1), we have
PPT Slide
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where ℕ is the set of positive integers. The Genocchi number Gn satisfy the recurrence relation
PPT Slide
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Therefore, we find out that G 2 = –1, G 4 = 1, G 6 = –3, G 8 = 17, G 10 = –155, G 12 = 2073, G 14 = –38227, .... That is, G 2n+1 = 0( n ≥ 1).
The Stirling number of the first kind s ( n, k ) can be defined by means of
PPT Slide
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or by the generating function
PPT Slide
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We get (1.9) from (1.7) and (1.8)
PPT Slide
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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
PPT Slide
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or by the generating function
PPT Slide
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We get (1.12) from (1.10) and (1.11)
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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
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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
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. First of all, we define the generalized Euler polynomials
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of the second kind as follows. In [7] , we introduced the generalized Euler polynomials
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of the second kind and investigate their properties. First of all, we introduce the generalized Euler polynomials
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of the second kind as follows. This completes with the usual convention of replacing
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(see, for details, [7] ).
Definition 2.1. Let x be a real or complex parameter, n k ( n, k ∈ ℕ). Then we define
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We derive that
PPT Slide
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where
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For a real and complex parameter α , the generalized Euler polynomials
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, 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
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By using Definition 2.2, we have the addition theorem of polynomials
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and the relation of polynomials
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and numbers
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Theorem 2.3. (Addition theorem) Let α, x, y ∈ ℂ and n be non-negative inte-gers. Then we get
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Proof . For n be non-negative integers, we have
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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
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where
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Proof . By (2.1) and (2.2), we easily have
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By Definition 2.2, (1.4) and (1.8) we have
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which readily yields
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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) = –3 α , ρ (α) (3, 2) = 0, ρ (α) (3, 3) = 1,
ρ (α) (4, 0) = 2 α +3 α 2 , ρ (α) (4, 1) = 0, ρ (α) (4, 2) = –6 α , ρ (α) (4, 3) = 0, ρ (α) (4, 4) = 1, . . .
Thus, we know that
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is a polynomial of x . Setting n = 1, 2, 3, 4, 5 in Theorem 2.4, we get to
PPT Slide
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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
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we have the following theorem.
Theorem 2.5. Let n, k ∈ ℕ, then by (1.1) and Definition 2.2, we have
PPT Slide
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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
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and Genocchi numbers Gn, we have
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where
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Proof . By (1.1), we have
PPT Slide
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Let us that
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Therefore, we have
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And we expand to degree of α , we obtain
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We deduce that by the generalized Euler polynomials
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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
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where
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Theorem 3.2. Let n k ( n, k ∈ ℕ). Then we obtain
PPT Slide
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Proof . By applying Theorem 2.4, we have
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It follows from Definition 2.2 that
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On the other hand, we have from (2.1)
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Substituting (3.9) in (3.8) we get
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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
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and Euler num-bers En, we have
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where
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Proof . By definition (1.1)
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Therefore, according to (3.1), (3.2), we have
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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
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Therefore, we have the following theorem.
Theorem 3.4. Let α be a real or complex parameter. Relation between
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and Euler polynomials
PPT Slide
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, we have
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For α = 1 in (3.13), we have the following corollary.
Corollary 3.5. For α = 1, we have
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Theorem 3.6. Let n, α ∈ ℕ. Relation between
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and Genocchi polynomi-als
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we have
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Proof . By Definition 2.2,
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Therefore, we may immediately obtain Theorem 3.6. This completes the proof of Theorem 3.6.
BIO
Y. H. Kim is a graduate student in Hannam University. Her research interests focus on functional analysis and special functions.
Department of Mathematics, Hannam University, Daejeon, 306-791, Korea
e-mail: sunbidori@hanmail.net
H. Y. Jung is a graduate student in Hannam University. His research interests focus on the scientific computing and functional analysis.
Department of Mathematics, Hannam University, Daejeon, 306-791, Korea
e-mail: tiger8049@korea.ac.kr
C. S. Ryoo received Ph.D. degree from Kyushu University. His research interests focus on the numerical verification method, scientific computing and p-adic functional analysis.
Department of Mathematics, Hannam University, Daejeon, 306-791, Korea
e-mail: ryoocs@hnu.kr
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