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SYMMETRIC PROPERTIES FOR GENERALIZED TWISTED q-EULER ZETA FUNCTIONS AND q-EULER POLYNOMIALS
SYMMETRIC PROPERTIES FOR GENERALIZED TWISTED q-EULER ZETA FUNCTIONS AND q-EULER POLYNOMIALS
Journal of Applied Mathematics & Informatics. 2016. Jan, 34(1_2): 107-114
Copyright © 2016, Korean Society of Computational and Applied Mathematics
  • Received : September 20, 2015
  • Accepted : November 11, 2015
  • Published : January 30, 2016
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About the Authors
N.S. JUNG
C.S. RYOO

Abstract
In this paper we give some symmetric property of the generalized twisted q -Euler zeta functions and q -Euler polynomials. AMS Mathematics Subject Classification : 11B68, 11S40, 11S80.
Keywords
1. Introduction
The Euler numbers and polynomials possess many interesting properties in many areas of mathematics and physics. Many mathematicians have studied in the area of various q -extensions of Euler polynomials and numbers (see [1 - 11] ). Recently, Y. Hu investigated several identities of symmetry for Carlitz's q -Bernoulli numbers and polynomials in complex field (see [3] ). D. Kim et al. [4] derived some identities of symmetry for Carlitz's q -Euler numbers and poly-nomials in complex field. J. Y. Kang and ℂ. S. Ryoo studied some identities of symmetry for q -Genocchi polynomials (see [2] ). In [1] , we obtained some iden-tities of symmetry for Carlitz's twisted q -Euler zeta function in complex field. In this paper, we establish some interesting symmetric identities for generalized twisted q -Euler zeta functions and generalized ] twisted q -Euler polynomials in complex field. If we take X = 1 in all equations of this article, then [1] are the special case of our results. Throughout this paper we use the following nota-tions. By ℕ we denote the set of natural numbers, ℤ denotes the ring of rational integers, ℚ denotes the field of rational numbers, ℂ denotes the set of complex numbers, and ℤ + = ℕ ∪ {0}. We use the following notation:
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Note that lim q→1 [ x ] = x . We assume that q ∈ ℂ with | q | < 1. Let r be a positive integer, and let ε be the r -th root of unity. Let X be Dirichlet's character with conductor d ∈ ℕ with d ≡ 1(mod2). Then the generalized twisted q -Euler polynomials associated with associated with X , En,X,q,ε , are defined by the following generating function
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and their values at x = 0 are called the generalized twisted q -Euler numbers and denoted En,X,q,ε .
By (1.1) and Cauchy product, we obtain
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with the usual convention about replacing ( EX,q,ε ) n by En,X,q,ε .
By using (1.1), we note that
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By (1.3), we are now ready to define the Hurwitz type of the generalized twisted q -Euler zeta functions.
Definition 1.1. Let s ∈ ℂ and x ∈ ℝ with x ≠ 0,−1,−2,.... We define
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Note that ζX,q,ε ( s, x ) is a meromorphic function on ℂ. Relation between ζX,q,ε ( s, x ) and Ek,X,q,ε ( x ) is given by the following theorem.
Theorem 1.2. For k ∈ ℕ, we get
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Observe that ζX,q,ε ( −k, x ) function interpolates Ek,X,q,ε ( x ) polynomials at non-negative integers. If X = 1, then ζX,q,ε ( s, x ) = ζq,ε ( s, x ) (see [1] ).
2. Symmetric property of generalized twistedq-Euler zeta functions
In this section, by using the similar method of [1 , 2 , 3 , 4 , 9] , expect for obvious modifications, we give some symmetric identities for generalized twisted q -Euler polynomials and generalized twisted q -Euler zeta functions. Let w 1 , w 2 ∈ ℕ with w 1 ≡ 1 (mod 2), w 2 ≡ 1 (mod 2).
Theorem 2.1. Let X be Dirichlet's character with conductor d ∈ ℕ with d ≡ 1 (mod 2) and ε be the r-th root of unity. For w 1 ,w 2 ∈ ℕ with w 1 ≡ 1 (mod 2) , w 2 ≡ 1 (mod 2) , we obtain
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Proof . Observe that [ xy ] q = [ x ] qy [ y ] q for any x, y ∈ ℂ. In Definition 1. 1, we derive next result by substitute
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for x in and replace q and ε by q w2 and ε w2 , respectively.
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Since for any non-negative integer n and odd positive integer w 1 , there exist unique non-negative integer r, j such that m = w 1 r + j with 0 ≤ j w 1 −1. So, the equation (2.1) can be written as
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In similarly, we obtain
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Using the method in (2.2), we obtain
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By (2.2) and (2.4), we obtain
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Next, we obtain the symmetric results by using definition and theorem of the generalized twisted q -Euler polynomials.
Theorem 2.2. Let X be Dirichlet's character with conductor d ∈ ℕ with d ≡ 1 (mod 2) and ε be the r-th root of unity. For w 1 , w 2 ∈ ℕ with w 1 ≡ 1 (mod 2) , w 2 ≡ 1 (mod 2) , i, j and n be non-negative integer, we obtain
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Proof . By substitute
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for x in Theorem 1. 2 and replace q and ε by q w2 and ε w2 , respectively, we derive
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Since for any non-negative integer m and odd positive integer w 1 , there exist unique non-negative integer r, j such that m = w 1 r + j with 0 ≤ j w 1 − 1.
Hence, the equation (2.6) is written as
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In similar, we obtain
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and
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It follows from the above equation that
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From (2.7), (2.8), (2.9) and (2.10), the proof of the Theorem 2.2 is completed. □
By (1.2) and Theorem 2.2, we have the following theorem.
Theorem 2.3. Let i, j and n be non-negative integers. For w 1 ,w 2 ∈ ℕ with w 1 ≡ 1 (mod 2), w 2 ≡ 1 (mod 2) , we have
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Proof . After some calculations, we have
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and
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By (2.11), (2.12) and Theorem 2. 2, we obtain that
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Hence, we have above theorem. □
By Theorem 2.3, we have the interesting symmetric identity for generalized twisted q -Euler numbers in complex field.
Corollary 2.4. For w 1 ,w 2 ∈ ℕ with w 1 ≡ 1 (mod 2) , w 2 ≡ 1 (mod 2) , we have
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BIO
N.S. Jung received Ph.D. degree from Hannam University. Her research interests are analytic number theory and p-adic functional analysis.
Department of Mathematics, Hannam University, Daejeon, 306-791, Korea.
e-mail: jns4235@nate.com
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
References
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Kang J.Y. , Ryoo C.S. (2014) On Symmetric Property for q-Genocchi Polynomials and Zeta Func-tion Int. Journal of Math. Analysis 8 9 - 16
He Yuan (2013) Symmetric identities for Carlitz's q-Bernoulli numbers and polynomials Advances in Difference Equations 246
Kim D. , Kim T. , Lee S.-H. , Seo J.-J. (2014) Symmetric Identities of the q-Euler Polynomials Adv. Studies Theor. Phys. 7 1149 - 1155
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