Multivariate p-Adic Fermionic q-Integral on Z p and Related Multiple Zeta-Type Functions

In 2008, Jang et al. constructed generating functions of the multiple twisted Carlitz’s type qBernoulli polynomials and obtained the distribution relation for them. They also raised the following problem: “are there analytic multiple twisted Carlitz’s type q-zeta functions which interpolate multiple twisted Carlitz’s type q-Euler (Bernoulli) polynomials?” The aim of this paper is to give a partial answer to this problem. Furthermore we derive some interesting identities related to twisted qextension of Euler polynomials and multiple twisted Carlitz’s type q-Euler polynomials.


Introduction, definitions, and notations
Let p be an odd prime.Z p , Q p , and C p will always denote, respectively, the ring of p-adic integers, the field of p-adic numbers, and the completion of the algebraic closure of Q p .Let v p : C p →Q ∪ {∞} Q is the field of rational numbers denote the p-adic valuation of C p normalized so that v p p 1. The absolute value on C p will be denoted as |•| p , and |x| p p −v p x for x ∈ C p .We let p is sometimes called a p-adic unit.For each integer N ≥ 0, C p N will denote the multiplicative group of the primitive p N th roots of unity in The dual of Z p , in the sense of p-adic Pontrjagin duality, is T p C p ∞ , the direct limit under inclusion of cyclic groups C p N of order p N N ≥ 0 , with the discrete topology.
When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ C p .If q ∈ C p , then we normally assume |1−q| p < p −1/ p−1 , so that q x exp x log q for |x| p ≤ 1.If q ∈ C, then we assume that |q| < 1. Let where a ∈ Z lies in 0 ≤ a < p N .We use the following notation: x q 1 − q x 1 − q . 1.3 Hence lim q→1 x q x 1.4 for any x with |x| p ≤ 1 in the present p-adic case.The distribution μ q a p N Z p is given as μ q a p N Z p q a p N q 1.5 cf.1-9 .For the ordinary p-adic distribution μ 0 defined by We say that f is a uniformly differentiable function at a point a ∈ Z p , we write f ∈ UD Z p , C p if the difference quotient has a limit f a as x, y → a, a .Also we use the following notation: x −q 1 − −q x 1 q , 1.9 cf.1-5 .
Min-Soo Kim et al. 3 In 1-3 , Kim gave a detailed proof of fermionic p-adic q-measures on Z p .He treated some interesting formulae-related q-extension of Euler numbers and polynomials; and he defined fermionic p-adic q-measures on Z p as follows: By using the fermionic p-adic q-measures, he defined the fermionic p-adic q-integral on Z p as follows: From 1.12 , we obtain where f 1 x f x 1 .By substituting f x e tx into 1.13 , classical Euler numbers are defined by means of the following generating function: These numbers are interpolated by the Euler zeta function which is defined as follows: cf. 1-9 .From 1.12 , we also obtain where f 1 x f x 1 .By substituting f x e tx into 1.13 , q-Euler numbers are defined by means of the following generating function: 1.17 These numbers are interpolated by the Euler q-zeta function which is defined as follows: In 6 , Ozden and Simsek defined generating function of q-Euler numbers by 2 q 1 Z p e tx dμ −q x 2 qe t 1 , 1.19 which are different from 1.17 .But we observe that all these generating functions were obtained by the same fermionic p-adic q-measures on Z p and the fermionic p-adic q-integrals on Z p .
In this paper, we define a multiple twisted Carlitz's type q-zeta functions, which interpolated multiple twisted Carlitz's type q-Euler polynomials at negative integers.This result gave us a partial answer of the problem proposed by Jang et al. 10 , which is given by: "Are there analytic multiple twisted Carlitz's type q-zeta functions which interpolate multiple twisted Carlitz's type q-Euler (Bernoulli) polynomials?"

Preliminaries
In 10 , Jang and Ryoo defined q-extension of Euler numbers and polynomials of higher order and studied multivariate q-Euler zeta functions.They also derived sums of products of q-Euler numbers and polynomials by using ferminonic p-adic q-integral.
In 5, 7 , Ozden et al. defined multivariate Barnes-type Hurwitz q-Euler zeta functions and l-functions.They also gave relation between multivariate Barnes-type Hurwitz q-Euler zeta functions and multivariate q-Euler l-functions.
In this section, we consider twisted q-extension of Euler numbers and polynomials of higher order and study multivariate twisted Barnes-type Hurwitz q-Euler zeta functions and l-functions.
Let UD Z h p , C p denote the space of all uniformly or strictly differentiable For a fixed positive integer d with d, p 1, we set 2 and 2.4 .Then we have where E h n,ω x are the twisted Euler polynomials of order h.From 2.5 , we note that We give an application of the multivariate q-deformed p-adic integral on Z h p in the fermionic sense related to 3 .Let

2.7
By substituting into 2.1 , we define twisted q-extension of Euler numbers of higher order by means of the following generating function: t n n! .

6
Abstract and Applied Analysis Then we have From 2.9 , we obtain where E h n,q,ω x is called twisted q-extension of Euler polynomials of higher order cf.11 .We note that if ω 1, then E h n,q,ω x E h n,q x and E h n,q,ω E h n,q cf.6 .We also note that The twisted q-extension of Euler polynomials of higher order, E h n,q,ω x , is defined by means of the following generating function: where |t log ωq | < π.From these generating functions of twisted q-extension of Euler polynomials of higher order, we construct twisted multiple q-Euler zeta functions as follows.
For s ∈ C and x ∈ R with 0 < x ≤ 1, we define

2.14
By the mth differentiation on both sides of 2.13 at t 0, we obtain the following for m 0, 1, . . . .From 2.14 and 2.15 , we arrive at the following x , m 0, 1, . . . .

2.16
We set

2.17
Let χ be Dirichlet's character with odd conductor d.We define twisted q-extension of generalized Euler polynomials of higher order by means of the following generating function cf.11 : Note that 2.20 8

Abstract and Applied Analysis
This allows us to rewrite 2.18 as

2.21
By applying the mth derivative operator d/dt m | t 0 in the above equation, we have for m 0, 1, . . . .From these generating functions of twisted q-extension of generalized Euler polynomials of higher order, we construct twisted multiple q-Euler l-functions as follows.For s ∈ C and x ∈ R with 0 < x ≤ 1, we define

2.24
Let s ∈ C and a i , F ∈ Z with F is an odd integer and 0 < a i < F, where i 1, . . ., h.Then twisted partial multiple q-Euler ζ-functions are as follows:

2.25
For i 1, . . ., h, substituting l i a i n i F with F is odd into 2.25 , we have

2.26
Then we obtain

2.28
Therefore, we modify twisted partial multiple q-Euler zeta functions as follows:

2.29
Let χ be a Dirichlet character with conductors d and d | F. From 2.23 and 2.27 , we have 2.30

The multiple twisted Carlitz's type q-Euler polynomials and q-zeta functions
Let us consider the multiple twisted Carlitz's type q-Euler polynomials as follows: cf. 1, 3 .These can be written as 3.2 We may now mention the following formulae which are easy to prove: x .

3.3
From 3.2 , we can derive generating function for the multiple twisted Carlitz's type q-Euler polynomials as follows:

3.4
Also, an obvious generating function for the multiple twisted Carlitz's type q-Euler polynomials is obtained, from 3.2 , by 2 h q e t/ 1−q n j 0 3.5 From now on, we assume that q ∈ C with |q| < 1.From 3.2 and 3.4 , we note that G z,h q,ω x, t

3.7
Thus we can define the multiple twisted Carlitz's type q-zeta functions as follows: In 12, Proposition 3 , Yamasaki showed that the series ζ z,h q,ω s, x converges absolutely for Re z > h − 1, and it can be analytically continued to the whole complex plane C. Note that if h 1, then −ω l q lz x l s q . 3.9 In 13 , Wakayama and Yamasaki studied q-analogue of the Hurwitz zeta function defined by the q-series with two complex variable s, z ∈ C: and special values at nonpositive integers of the q-analogue of the Hurwitz zeta function.Therefore, by the mth differentiation on both sides of 3.6 at t 0, we obtain the following: 3.12 for m 0, 1, . . . .From 3.7 , 3.8 , and 3.12 , we have 3.13 which shows that the multiple twisted Carlitz's type q-zeta functions interpolate the multiple twisted Carlitz's type q-Euler numbers and polynomials.For m 0, 1, . . ., we have where x ∈ R and 0 < x ≤ 1.Thus, we derive the analytic multiple twisted Carlitz's type q-zeta functions which interpolate multiple twisted Carlitz's type q-Euler polynomials.This gives a part of the answer to the question proposed in 10 .

Remarks
For nonnegative integers m and n, we define the q-binomial coefficient m n q by m n q q; q m q; q n q; q m−n , 4.1 where a; q m m−1 k 0 1 − aq k for m ≥ 1 and a; q 0 1.For h ∈ N, it holds that −ω l q z 1 l l x s q l 1 ,...,l h ≥0 −ω l q z−h 1 l l x s q .

4.3
We set m q !m q m − 1 q • • • 1 q for m ∈ N. The following identity has been studied in 12 : l x q − q l j x − j q h−1 k 0 q l h−1−k P k q,h x l x k q , 4.4 where P k q,h x , 0 ≤ k ≤ h − 1, is a function of x defined by for 0 ≤ k ≤ h − 2 and P h−1 q,h x 1/ h − 1 q !.By using 3.9 , 4.3 , and 4.5 , we have and so The values of ζ z,h q,ω −m, x at h 2, 3 are given explicitly as follows: 4.8 2