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SYMMETRIC IDENTITIES FOR TWISTED q-EULER ZETA FUNCTIONS
SYMMETRIC IDENTITIES FOR TWISTED q-EULER ZETA FUNCTIONS
Journal of Applied Mathematics & Informatics. 2015. Sep, 33(5_6): 649-656
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : May 25, 2015
  • Accepted : August 03, 2015
  • Published : September 30, 2015
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About the Authors
N.S. JUNG
C.S. RYOO

Abstract
In this paper we investigate some symmetric property of the twisted q -Euler zeta functions and twisted q -Euler polynomials. AMS Mathematics Subject Classification : 11B68, 11S40, 11S80.
Keywords
1. Introduction
The Euler polynomials and numbers possess many interesting properties in many areas of mathematics and physics. Many mathematicians have studied in the area of the q -extension of the Euler numbers and polynomials (see [3 - 10] ).
Recently, Y. Hu studied several identities of symmetry for Carlitz's q -Bernoulli numbers and polynomials in complex field (see [2] ). D. Kim et al. [3] derived some identities of symmetry for Carlitz's q -Euler numbers and polynomials in complex field. J.Y. Kang and C.S. Ryoo investigated some identities of symmetry for q -Genocchi polynomials (see [1] ). In [8] , we obtained some identities of symmetry for Carlitz's twisted q -Euler polynomials associated with p -adic q -integral on ℤ p . In this paper, we establish some interesting symmetric identities for twisted q -Euler zeta functions and twisted q -Euler polynomials in complex field. If we take ε = 1 in all equations of this article, then [3] are the special case of our results. Throughout this paper we use the following notations. 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 ε be the pN -th root of unity. Then the twisted q -Euler polynomials En,q,ε are defined by the generating function to be
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When x = 0, En,q,ε = En,q,ε (0) are called the twisted q -Euler numbers. By (1.1) and Cauchy product, we have
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with the usual convention about replacing ( Eq,ε ) n by En,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 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 ζq,ζ ( s , x ) is a meromorphic function on ℂ. A relation between ζq,ε ( s , x ) and Ek,q,ε ( x ) is given by the following theorem.
Theorem 1.2. For k ∈ ℕ, we get
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Observe that ζq,ε (− k , x ) function interpolates Ek,q,ε ( x ) polynomials at non-negative integers.
2. Symmetric property of twistedq-Euler zeta functions
In this section, by using the similar method of [1 , 2 , 3] , expect for obvi-ous modifications, we investigate some symmetric identities for twisted q -Euler polynomials and 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. For w 1 , w 2 ∈ ℕ with w 1 ≡ 1 (mod 2), w 2 ≡ 1 (mod 2), we have
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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 can see that
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Using the method in (2.2), we obtain
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From (2.2) and (2.4), we have
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Next, we derive the symmetric results by using definition and theorem of the twisted q -Euler polynomials.
Theorem 2.2. 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 . 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 have
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and
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It follows from the above equation that
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From (2.8) and (2.9), 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 obtain
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and
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From (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 obtain the interesting symmetric identity for 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
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
Kim T. (2007) q-Euler numbers and polynomials associated with p-adic q-integrals J. Nonlinear Math. Phys. 14 15 - 27    DOI : 10.2991/jnmp.2007.14.1.3
Kupershmidt B.A. (2005) Reflection symmetries of q-Bernoulli polynomials J. Nonlinear Math. Phys. 12 412 - 422    DOI : 10.2991/jnmp.2005.12.s1.34
Ryoo C.S. On the Barnes type multiple q-Euler polynomials twisted by ramified roots of unity Proc. Jangjeon Math. Soc. 13 2010 255 - 263
Ryoo C.S. (2011) A note on the weighted q-Euler numbers and polynomials Advan. Stud. Contemp. Math. 21 47 - 54
Ryoo C.S. (2015) Some Identities of Symmetry for Carlitz's Twisted q-Euler Polynomials As-sociated with p-Adic q-Integral on Zp Int. Journal of Math. Analysis 9 1747 - 1753
Ryoo C.S. (2014) Analytic Continuation of Euler Polynomials and the Euler Zeta Function Discrete Dynamics in Nature and Society Article ID 568129 2014    DOI : 10.1155/2014/568129
Ryoo C.S. (2009) A Note on the Reflection Symmetries of the Genocchi polynomials J. Appl. Math. & Informatics 27 1397 - 1404