SYMMETRIC PROPERTIES FOR GENERALIZED TWISTED q-EULER ZETA FUNCTIONS AND q-EULER POLYNOMIALS

Abstract

In this paper we give some symmetric property of the generalized twisted q-Euler zeta functions and q-Euler polynomials.

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:

Note that limq→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

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

with the usual convention about replacing (EX,q,ε)n by En,X,q,ε.

By using (1.1), we note that

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

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

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 twisted q-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 w1,w2 ∈ ℕ with w1 ≡ 1 (mod 2), w2 ≡ 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 w1,w2 ∈ ℕ with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2), we obtain

Proof. Observe that [xy]q = [x]qy [y]q for any x, y ∈ ℂ. In Definition 1. 1, we derive next result by substitute for x in and replace q and ε by qw2 and εw2 , respectively.

Since for any non-negative integer n and odd positive integer w1, there exist unique non-negative integer r, j such that m = w1r+j with 0 ≤ j ≤ w1 −1. So, the equation (2.1) can be written as

In similarly, we obtain

Using the method in (2.2), we obtain

By (2.2) and (2.4), we obtain

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 w1,w2 ∈ ℕ with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2), i, j and n be non-negative integer, we obtain

Proof. By substitute for x in Theorem 1. 2 and replace q and ε by qw2 and εw2, respectively, we derive

Since for any non-negative integer m and odd positive integer w1, there exist unique non-negative integer r, j such that m = w1r + j with 0 ≤ j ≤ w1 − 1.

Hence, the equation (2.6) is written as

In similar, we obtain

and

It follows from the above equation that

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 w1,w2 ∈ ℕ with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2), we have

Proof. After some calculations, we have

and

By (2.11), (2.12) and Theorem 2. 2, we obtain that

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 w1,w2 ∈ ℕ with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2), we have