SYMMETRIC IDENTITIES FOR TWISTED q-EULER ZETA FUNCTIONS

Abstract

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

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 ℤp. 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:

Note that limq→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

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

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

By using (1.1), we note that

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

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

Observe that ζq,ε(−k, x) function interpolates Ek,q,ε(x) polynomials at non-negative integers.

 

2. Symmetric property of twisted q-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 w1,w2 ∈ ℕ with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2).

Theorem 2.1. For w1,w2 ∈ ℕ with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2), we have

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 can see that

Using the method in (2.2), we obtain

From (2.2) and (2.4), we have

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

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 have

and

It follows from the above equation that

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

Proof. After some calculations, we obtain

and

From (2.11), (2.12) and Theorem 2.2, we obtain that

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