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

Abstract

In 2008, Jang et al. constructed generating functions of the multiple twisted Carlitz's type 𝑞-Bernoulli 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 𝑞-extension of Euler polynomials and multiple twisted Carlitz's type 𝑞-Euler polynomials.

1. 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^{−𝑣𝑝(𝑥)} for 𝑥∈ℂ_𝑝. We let ℤ^×_𝑝={𝑥∈ℤ_𝑝∣1/𝑥∈ℤ_𝑝}. A 𝑝-adic integer in ℤ^×_𝑝 is sometimes called a 𝑝-adic unit. For each integer 𝑁≥0, 𝐶_𝑝𝑁 will denote the multiplicative group of the primitive 𝑝^𝑁th roots of unity in ℂ^×_𝑝=ℂ_𝑝⧵{0}. Set𝐓_𝑝={𝜔∈ℂ_𝑝∣𝜔^𝑝𝑁=1forsome𝑁≥0}=_𝑁≥0𝐶_𝑝𝑁. (1.1)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_𝑝𝑁 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(𝑥log𝑞) for |𝑥|_𝑝≤1. If 𝑞∈ℂ, then we assume that |𝑞|<1.

Letℤ_𝑝=lim_←𝑁(ℤ𝑝^𝑁ℤ),ℤ^×_𝑝=_0<𝑎<𝑝𝑎+𝑝ℤ_𝑝,𝑎+𝑝^𝑁ℤ_𝑝={𝑥∈ℤ_𝑝∣𝑥≡𝑎(mod𝑝^𝑁)}, (1.2)where 𝑎∈ℤ lies in 0≤𝑎<𝑝^𝑁.

We use the following notation:[𝑥]_𝑞=1−𝑞^𝑥.1−𝑞 (1.3)Hencelim_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^𝑁ℤ_𝑝) is given as𝜇_𝑞(𝑎+𝑝^𝑁ℤ_𝑝𝑞)=^𝑎[𝑝^𝑁]_𝑞 (1.5)(cf. [1–9]). For the ordinary 𝑝-adic distribution 𝜇₀ defined by𝜇₀(𝑎+𝑝^𝑁ℤ_𝑝1)=𝑝^𝑁, (1.6)we seelim_𝑞→1𝜇_𝑞=𝜇₀. (1.7)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 quotient𝐹_𝑓(𝑥,𝑦)=𝑓(𝑥)−𝑓(𝑦)𝑥−𝑦 (1.8)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]).

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:𝜇_−𝑞(𝑎+𝑝^𝑁ℤ_𝑝)=(−𝑞)^𝑎[𝑝^𝑁]_−𝑞. (1.10)By using the fermionic 𝑝-adic 𝑞-measures, he defined the fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 as follows:𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇_−𝑞(𝑥)=lim_𝑁→∞1[𝑝^𝑁]^𝑝_−𝑞^𝑁−1_𝑥=0𝑓(𝑥)(−𝑞)^𝑥 (1.11)for 𝑓∈UD(ℤ_𝑝,ℂ_𝑝) (cf. [1–3]). Observe that𝐼₋₁(𝑓)=lim_𝑞→1𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝑥)𝑑𝜇₋₁(𝑥)=lim^𝑝_𝑁→∞^𝑁−1_𝑥=0𝑓(𝑥)(−1)^𝑥. (1.12)From (1.12), we obtain𝐼₋₁(𝑓₁)+𝐼₋₁(𝑓)=2𝑓(0), (1.13)where 𝑓₁(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:_ℤ𝑝𝑒^𝑡𝑥𝑑𝜇₋₁2(𝑥)=𝑒^𝑡=+1^∞_𝑛=0𝐸_𝑛𝑡^𝑛.𝑛! (1.14)These numbers are interpolated by the Euler zeta function which is defined as follows:𝜁_𝐸(𝑠)=^∞_𝑛=1(−1)^𝑛𝑛^𝑠,𝑠∈ℂ, (1.15)(cf. [1–9]). From (1.12), we also obtain𝑞𝐼_−𝑞(𝑓₁)+𝐼_−𝑞(𝑓)=[2]_𝑞𝑓(0), (1.16)where 𝑓₁(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:_ℤ𝑝𝑒^𝑡𝑥𝑑𝜇_−𝑞(𝑥)=[2]_𝑞𝑞𝑒^𝑡=+1^∞_𝑛=0𝐸_𝑛,𝑞𝑡^𝑛.𝑛! (1.17)These numbers are interpolated by the Euler 𝑞-zeta function which is defined as follows:𝜁_𝑞,𝐸(𝑠)=[2]_𝑞^∞_𝑛=1(−1)^𝑛𝑞^𝑛𝑛^𝑠,𝑠∈ℂ, (1.18)(cf. [4]).

In [6], Ozden and Simsek defined generating function of 𝑞-Euler numbers by2𝑞+1_ℤ𝑝𝑒^𝑡𝑥𝑑𝜇_−𝑞2(𝑥)=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?”

2. 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 ℂ_𝑝-valued functions on ℤ^ℎ_𝑝=ℤ_𝑝×⋯×ℤ_{𝑝ℎ-times}. For 𝑓∈UD(ℤ^ℎ_𝑝,ℂ_𝑝), the 𝑝-adic 𝑞-integral on ℤ^ℎ_𝑝 is defined by𝐼^(ℎ)_−𝑞(𝑓)=_ℤ𝑝⋯_{ℤ𝑝ℎ-times}𝑓(𝑥₁,…,𝑥_ℎ)𝑑𝜇_−𝑞(𝑥₁)⋯𝑑𝜇_−𝑞(𝑥_ℎ)=lim_𝑁→∞1[𝑝^𝑁]^ℎ𝑝_−𝑞^𝑁−1_𝑥1=0⋯^𝑝𝑁−1_𝑥ℎ=0𝑓(𝑥₁,…,𝑥_ℎ)(−𝑞)^{𝑥1+⋯+𝑥ℎ} (2.1)(cf. [3]). If 𝑞→1, then𝐼^(ℎ)₋₁(𝑓)=lim_𝑞→1𝐼^(ℎ)_−𝑞(𝑓)=lim^𝑝_𝑁→∞^𝑁−1_𝑥1=0⋯^𝑝𝑁−1_𝑥ℎ=0𝑓(𝑥₁,…,𝑥_ℎ)(−1)^{𝑥1+⋯+𝑥ℎ}. (2.2)For a fixed positive integer 𝑑 with (𝑑,𝑝)=1, we set𝑋_𝑝=lim_←𝑁(ℤ𝑑𝑝^𝑁ℤ). (2.3)For 𝑓∈UD(ℤ^ℎ_𝑝,ℂ_𝑝),𝐼^(ℎ)₋₁(𝑓)=_𝑋𝑝⋯_{𝑋𝑝ℎ-times}𝑓(𝑥₁,…,𝑥_ℎ)𝑑𝜇₋₁(𝑥₁)⋯𝑑𝜇₋₁(𝑥_ℎ), (2.4)(cf. [2]).

We set 𝑓(𝑥₁,…,𝑥_ℎ)=𝜔^{𝑥1+⋯+𝑥ℎ}𝑒^{(𝑥+𝑥1+⋯+𝑥ℎ)𝑡} in (2.2) and (2.4). Then we have_{ℤ𝑝ℤ𝑝}⋯_ℎ-times𝜔^{𝑥1+⋯+𝑥ℎ}𝑒^{(𝑥+𝑥1+⋯+𝑥ℎ)𝑡}𝑑𝜇₋₁(𝑥₁)⋯𝑑𝜇₋₁(𝑥_ℎ2)=(𝜔𝑒^𝑡2+1)⋯(𝜔𝑒^𝑡)+1_ℎ-times=^∞_𝑛=0𝐸^(ℎ)_𝑛,𝜔𝑡(𝑥)^𝑛,𝑛! (2.5)where 𝐸^(ℎ)_𝑛,𝜔(𝑥) are the twisted Euler polynomials of order ℎ. From (2.5), we note that_{ℤ𝑝ℤ𝑝}⋯_ℎ-times𝜔^{𝑥1+⋯+𝑥ℎ}(𝑥+𝑥₁+⋯+𝑥_ℎ)^𝑛𝑑𝜇₋₁(𝑥₁)⋯𝑑𝜇₋₁(𝑥_ℎ)=𝐸^(ℎ)_𝑛,𝜔(𝑥). (2.6)

We give an application of the multivariate 𝑞-deformed 𝑝-adic integral on ℤ^ℎ_𝑝 in the fermionic sense related to [3]. Let_ℤℎ𝑝=_ℤ𝑝⋯_{ℤ𝑝ℎ-times}. (2.7)By substituting𝑓(𝑥₁,…,𝑥_ℎ)=𝜔^{𝑥1+⋯+𝑥ℎ}𝑒^{(𝑥+𝑥1+⋯+𝑥ℎ)𝑡} (2.8)into (2.1), we define twisted 𝑞-extension of Euler numbers of higher order by means of the following generating function:_ℤℎ𝑝𝜔^{𝑥1+⋯+𝑥ℎ}𝑒^{(𝑥1+⋯+𝑥ℎ)𝑡}𝑑𝜇_−𝑞(𝑥₁)⋯𝑑𝜇_−𝑞(𝑥_ℎ)=([2]_𝑞𝜔𝑞𝑒^𝑡+1)⋯([2]_𝑞𝜔𝑞𝑒^𝑡)+1_ℎ-times=^∞_𝑛=0𝐸^(ℎ)_{𝑛,𝑞,𝜔}𝑡^𝑛.𝑛! (2.9)Then we have_ℤℎ𝑝𝜔^{𝑥1+⋯+𝑥ℎ}(𝑥₁+⋯+𝑥_ℎ)^𝑛𝑑𝜇_−𝑞(𝑥₁)⋯𝑑𝜇_−𝑞(𝑥_ℎ)=𝐸^(ℎ)_{𝑛,𝑞,𝜔}. (2.10)From (2.9), we obtain_ℤℎ𝑝𝜔^{𝑥1+⋯+𝑥ℎ}𝑒^{(𝑥+𝑥1+⋯+𝑥ℎ)𝑡}𝑑𝜇_−𝑞(𝑥₁)⋯𝑑𝜇_−𝑞(𝑥_ℎ)=[2]^ℎ_𝑞𝑒^𝑥𝑡(𝜔𝑞𝑒^𝑡+1)⋯(𝜔𝑞𝑒^𝑡+1)_ℎ-times=^∞_𝑛=0𝐸^(ℎ)_{𝑛,𝑞,𝜔}𝑡(𝑥)^𝑛,𝑛! (2.11)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 that𝐸^(ℎ)_{𝑛,𝑞,𝜔}(𝑥)=^𝑛_𝑘=0𝑛𝑘𝐸^(ℎ)_{𝑘,𝑞,𝜔}𝑥^𝑛−𝑘. (2.12)

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:𝐺^(ℎ)_𝑞,𝜔(𝑥,𝑡)=[2]_𝑞𝜔𝑞𝑒^𝑡⋯+1[2]_𝑞𝜔𝑞𝑒^𝑡+1_ℎ-times𝑒^𝑥𝑡=[2]^ℎ_𝑞𝑒^∞𝑡𝑥_𝑙1=0(−𝜔)^𝑙1𝑞^𝑙1𝑒^𝑙1𝑡⋯^∞_𝑙ℎ=0(−𝜔)^𝑙ℎ𝑞^𝑙ℎ𝑒^𝑙ℎ𝑡=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜔)^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1+⋯+𝑙ℎ}𝑒^{(𝑙1+⋯+𝑙ℎ+𝑥)𝑡}=^∞_𝑛=0𝐸^(ℎ)_{𝑛,𝑞,𝜔}𝑡(𝑥)^𝑛,𝑛! (2.13)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 define𝜁^(ℎ)_{𝑞,𝜔,𝐸}(𝑠,𝑥)=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜔)^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1+⋯+𝑙ℎ}(𝑙₁+⋯+𝑙_ℎ+𝑥)^𝑠. (2.14)By the 𝑚th differentiation on both sides of (2.13) at 𝑡=0, we obtain the following𝐸^(ℎ)_{𝑚,𝑞,𝜔}(𝑥)=(dd𝑡)^𝑚𝐺^(ℎ)_𝑞,𝜔(𝑥,𝑡)|_𝑡=0=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜔)^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1+⋯+𝑙ℎ}(𝑥+𝑙₁+⋯+𝑙_ℎ)^𝑚 (2.15)for 𝑚=0,1,….

From (2.14) and (2.15), we arrive at the following𝜁^(ℎ)_{𝑞,𝜔,𝐸}(−𝑚,𝑥)=𝐸^(ℎ)_{𝑚,𝑞,𝜔}(𝑥),𝑚=0,1,…. (2.16)

We set_𝑋ℎ𝑝=_𝑋𝑝⋯_{𝑋𝑝ℎ-times}. (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]):_𝑋ℎ𝑝𝜒(𝑥₁+⋯+𝑥_ℎ)𝜔^{𝑥1+⋯+𝑥ℎ}𝑒^{(𝑥+𝑥1+⋯+𝑥ℎ)𝑡}𝑑𝜇_−𝑞(𝑥₁)⋯𝑑𝜇_−𝑞(𝑥_ℎ)=^∞_𝑛=0𝐸^(ℎ)_{𝑛,𝑞,𝜔,𝜒}𝑡(𝑥)^𝑛.𝑛! (2.18)Note that^∞_𝑛=0𝐸^(ℎ)_{𝑛,𝑞,𝜔,𝜒}𝑡(𝑥)^𝑛𝑛!=𝑒^𝑥𝑡_𝑋𝑝⋯_{𝑋𝑝ℎ-times}𝜒(𝑥₁+⋯+𝑥_ℎ)𝜔^{𝑥1+⋯+𝑥ℎ}𝑒^{(𝑥1+⋯+𝑥ℎ)𝑡}𝑑𝜇_−𝑞(𝑥₁)⋯𝑑𝜇_−𝑞(𝑥_ℎ)=𝑒^𝑥𝑡1[𝑑]^ℎ_−𝑞lim_𝑁→∞1[𝑝^𝑁]_(−𝑞)^𝑑𝑑−1_𝑎1^𝑝₌₀^𝑁−1_𝑥1=0⋯^𝑑−1_𝑎ℎ^𝑝₌₀^𝑁−1_𝑥ℎ=0𝜒(𝑎₁+𝑑𝑥₁+⋯+𝑎_ℎ+𝑑𝑥_ℎ)×𝜔^{𝑎1+𝑑𝑥1+⋯+𝑎ℎ+𝑑𝑥ℎ}𝑒^{(𝑎1+𝑑𝑥1+⋯+𝑎ℎ+𝑑𝑥ℎ)𝑡}(−𝑞)^{𝑎1+𝑑𝑥1+⋯+𝑎ℎ+𝑑𝑥ℎ}=𝑒^𝑥𝑡1[𝑑]^ℎ_−𝑞^𝑑−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)𝜔^{𝑎1+⋯+𝑎ℎ}(−𝑞)^{𝑎1+⋯+𝑎ℎ}𝑒^{(𝑎1+⋯+𝑎ℎ)𝑡}×lim_𝑁→∞1+𝑞^𝑑1+𝑞^𝑑𝑝𝑁1+𝜔^𝑑𝑝𝑁𝑞^𝑑𝑝𝑁𝑒^𝑑𝑝𝑁1+𝜔^𝑑𝑞^𝑑𝑒^𝑑𝑡⋯lim_𝑁→∞1+𝑞^𝑑1+𝑞^𝑑𝑝𝑁1+𝜔^𝑑𝑝𝑁𝑞^𝑑𝑝𝑁𝑒^𝑑𝑝𝑁1+𝜔^𝑑𝑞^𝑑𝑒^𝑑𝑡_ℎ−times=𝑒^𝑥𝑡1[𝑑]^ℎ_−𝑞^𝑑−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)𝜔^{𝑎1+⋯+𝑎ℎ}(−𝑞)^{𝑎1+⋯+𝑎ℎ}𝑒^{(𝑎1+⋯+𝑎ℎ)𝑡}×1+𝑞^𝑑1+𝜔^𝑑𝑞^𝑑𝑒^𝑑𝑡⋯1+𝑞^𝑑1+𝜔^𝑑𝑞^𝑑𝑒^𝑑𝑡_ℎ-times (2.19)sincelim_𝑁→∞𝑞^𝑝𝑁=1for|1−𝑞|_𝑝<1. (2.20) This allows us to rewrite (2.18) as^∞_𝑛=0𝐸^(ℎ)_{𝑛,𝑞,𝜔,𝜒}𝑡(𝑥)^𝑛𝑛!=𝑒^𝑥𝑡1[𝑑]^ℎ_−𝑞^𝑑−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)𝜔^{𝑎1+⋯+𝑎ℎ}(−𝑞)^{𝑎1+⋯+𝑎ℎ}𝑒^{(𝑎1+⋯+𝑎ℎ)𝑡}×1+𝑞^𝑑1+𝜔^𝑑𝑞^𝑑𝑒^𝑑𝑡⋯1+𝑞^𝑑1+𝜔^𝑑𝑞^𝑑𝑒^𝑑𝑡_ℎ-times=[2]^ℎ_𝑞𝑒^{𝑥𝑡𝑑−1}_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)𝜔^{𝑎1+⋯+𝑎ℎ}(−𝑞)^{𝑎1+⋯+𝑎ℎ}𝑒^{(𝑎1+⋯+𝑎ℎ)𝑡}×^∞_𝑥1=0(−𝜔^𝑑𝑞^𝑑𝑒^𝑑𝑡)^𝑥1⋯^∞_𝑥ℎ=0(−𝜔^𝑑𝑞^𝑑𝑒^𝑑𝑡)^𝑥ℎ_ℎ-times=[2]^ℎ_𝑞𝑒^∞𝑥𝑡_{𝑥1,…,𝑥ℎ=0}^𝑑−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+𝑑𝑥₁+⋯+𝑎_ℎ+𝑑𝑥_ℎ)×𝜔^{𝑎1+𝑑𝑥1+⋯+𝑎ℎ+𝑑𝑥ℎ}(−𝑞)^{𝑎1+𝑑𝑥1+⋯+𝑎ℎ+𝑑𝑥ℎ}𝑒^{(𝑎1+𝑑𝑥1+⋯+𝑎ℎ+𝑑𝑥ℎ)𝑡}=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−1)^{𝑙1+⋯+𝑙ℎ}𝜒(𝑙₁+⋯+𝑙_ℎ)𝜔^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1+⋯+𝑙ℎ}𝑒^{(𝑥+𝑙1+⋯+𝑙ℎ)𝑡}. (2.21)By applying the 𝑚th derivative operator (d/d𝑡)^𝑚|_𝑡=0 in the above equation, we have𝐸^(ℎ)_{𝑚,𝑞,𝜔,𝜒}(𝑥)=[2]∑^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}𝜒(𝑙₁+⋯+𝑙_ℎ)∏^ℎ_𝑖=1(−1)^𝑙𝑖𝜔^𝑙𝑖𝑞^𝑙𝑖(𝑥+𝑙₁+⋯+𝑙_ℎ)^𝑚 (2.22)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 define𝑙^(ℎ)_{𝑞,𝜔,𝐸}(𝑠,𝑥,𝜒)=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}𝜒(𝑙₁+⋯+𝑙_ℎ)∏^ℎ_𝑖=1(−1)^𝑙𝑖𝜔^𝑙𝑖𝑞^𝑙𝑖(𝑙₁+⋯+𝑙_ℎ+𝑥)^𝑠. (2.23)From (2.22) and (2.23), we arrive at the following𝑙^(ℎ)_{𝑞,𝜔,𝐸}(−𝑚,𝑥,𝜒)=𝐸^(ℎ)_{𝑚,𝑞,𝜔,𝜒}(𝑥),𝑚=0,1,…. (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:𝐻^(ℎ)_{𝑞,𝜔,𝐸}(𝑠,𝑎₁,…,𝑎_ℎ,𝑥∣𝐹)=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ𝑙=0𝑖≡𝑎𝑖(mod𝐹),𝑖=1,…,ℎ}(−1)^{𝑙1+⋯+𝑙ℎ}𝜔^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1+⋯+𝑙ℎ}(𝑙₁+⋯+𝑙_ℎ+𝑥)^𝑠. (2.25)For 𝑖=1,…,ℎ, substituting 𝑙_𝑖=𝑎_𝑖+𝑛_𝑖𝐹 with 𝐹 is odd into (2.25), we have𝐻^(ℎ)_{𝑞,𝜔,𝐸}(𝑠,𝑎₁,…,𝑎_ℎ,𝑥∣𝐹)=[2]^ℎ_𝑞^∞_{𝑛1,…,𝑛ℎ=0}(−1)^{𝑎1+𝑛1𝐹+⋯+𝑎ℎ+𝑛ℎ𝐹}𝜔^{𝑎1+𝑛1𝐹+⋯+𝑎ℎ+𝑛ℎ𝐹}𝑞^{𝑎1+𝑛1𝐹+⋯+𝑎ℎ+𝑛ℎ𝐹}(𝑎₁+𝑛₁𝐹+⋯+𝑎_ℎ+𝑛_ℎ𝐹+𝑥)^𝑠=[2]^ℎ_𝑞[2]^ℎ_𝑞𝐹(−𝜔𝑞)^{𝑎1+⋯+𝑎ℎ}𝐹^𝑠[2]^ℎ_𝑞𝐹^∞_{𝑛1,…,𝑛ℎ=0}(−1)^{𝑛1+⋯+𝑛ℎ}(𝜔^𝐹)^{𝑛1+⋯+𝑛ℎ}(𝑞^𝐹)^{𝑛1+⋯+𝑛ℎ}(𝑛₁+⋯+𝑛_ℎ+(𝑎₁+⋯+𝑎_ℎ+𝑥)/𝐹)^𝑠=[2]^ℎ_𝑞[2]^ℎ_𝑞𝐹(−𝜔𝑞)^{𝑎1+⋯+𝑎ℎ}𝐹^𝑠𝜁_𝑞^(ℎ)_{𝐹,𝜔𝐹,𝐸}𝑎(𝑠,¹+⋯+𝑎_ℎ+𝑥𝐹). (2.26)Then we obtain𝐻^(ℎ)_{𝑞,𝜔,𝐸}(𝑠,𝑎₁,…,𝑎_ℎ,𝑥∣𝐹)=[2]^ℎ_𝑞[2]^ℎ_𝑞𝐹(−𝜔𝑞)^{𝑎1+⋯+𝑎ℎ}𝐹^𝑠𝜁_𝑞^(ℎ)_{𝐹,𝜔𝐹,𝐸}𝑎(𝑠,¹+⋯+𝑎_ℎ+𝑥𝐹). (2.27)By using (2.12) and (2.27) and by substituting 𝑠=−𝑚,𝑚=0,1,…, we get𝐻^(ℎ)_{𝑞,𝜔,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ,𝑥∣𝐹)=[2]^ℎ_𝑞[2]^ℎ_𝑞𝐹(−𝜔𝑞)^{𝑎1+⋯+𝑎ℎ}(𝑎₁+⋯+𝑎_ℎ+𝑥)^𝑚×^𝑚_𝑘=0𝑚𝑘(𝐹𝑎₁+⋯+𝑎_ℎ)+𝑥^𝑘𝐸^(ℎ)_{𝑘,𝑞𝐹,𝜔𝐹}. (2.28)Therefore, we modify twisted partial multiple 𝑞-Euler zeta functions as follows:𝐻^(ℎ)_{𝑞,𝜔,𝐸}(𝑠,𝑎₁,…,𝑎_ℎ,𝑥∣𝐹)=[2]^ℎ_𝑞[2]^ℎ_𝑞𝐹(−𝜔𝑞)^{𝑎1+⋯+𝑎ℎ}(𝑎₁+⋯+𝑎_ℎ+𝑥)^−𝑠×^∞_𝑘=0𝑘(−𝑠𝐹𝑎₁+⋯+𝑎_ℎ)+𝑥^𝑘𝐸^(ℎ)_{𝑘,𝑞𝐹,𝜔𝐹}. (2.29)Let 𝜒 be a Dirichlet character with conductors 𝑑 and 𝑑∣𝐹. From (2.23) and (2.27), we have𝑙^(ℎ)_{𝑞,𝜔,𝐸}(𝑠,𝑥,𝜒)=[2]^ℎ_𝑞[2]^ℎ_𝑞𝐹𝐹^{−𝑠𝐹−1}_{𝑎1,…,𝑎ℎ=0}(−𝜔𝑞)^{𝑎1+⋯+𝑎ℎ}×𝜒(𝑎₁+⋯+𝑎_ℎ)𝜁_𝑞^(ℎ)_{𝐹,𝜔𝐹,𝐸}𝑎(𝑠,¹+⋯+𝑎_ℎ+𝑥𝐹)=^𝐹−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)𝐻^(ℎ)_{𝑞,𝜔,𝐸}(𝑠,𝑥,𝑎₁,…,𝑎_ℎ,𝑥∣𝐹). (2.30)

3. 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:𝐸^(𝑧,ℎ)_{𝑛,𝑞,𝜔}(𝑥)=_ℤ𝑝⋯_{ℤ𝑝ℎ-times}[𝑥₁+⋯+𝑥_ℎ+𝑥]^𝑛_𝑞𝜔^{𝑥1+⋯+𝑥ℎ}𝑞^{𝑥1(𝑧−1)+⋯+𝑥ℎ(𝑧−ℎ)}𝑑𝜇_𝑞(𝑥₁)⋯𝑑𝜇_𝑞(𝑥_ℎ) (3.1)(cf. [1, 3]). These can be written as𝐸^(𝑧,ℎ)_{𝑛,𝑞,𝜔}(𝑥)=[2]^ℎ_𝑞(1−𝑞)^𝑛𝑛_𝑖=0𝑛𝑖𝑞^𝑖𝑥(−1)^𝑖11+𝜔𝑞^𝑧+𝑖⋯11+𝜔𝑞^{𝑧+𝑖−ℎ+1}. (3.2)We may now mention the following formulae which are easy to prove:𝜔𝑞^𝑧𝐸^(𝑧,ℎ)_{𝑛,𝑞,𝜔}(𝑥+1)+𝐸^(𝑧,ℎ)_{𝑛,𝑞,𝜔}(𝑥)=[2]_𝑞𝐸^{(𝑧−1,ℎ−1)}_{𝑛,𝑞,𝜔}(𝑥). (3.3)From (3.2), we can derive generating function for the multiple twisted Carlitz's type 𝑞-Euler polynomials as follows:^∞_𝑛=0𝐸^(𝑧,ℎ)_{𝑛,𝑞,𝜔}𝑡(𝑥)^𝑛=𝑛!_∞𝑛=0[2]^ℎ_𝑞(1−𝑞)^𝑛𝑛_𝑖=0𝑛𝑖𝑞^𝑖𝑥(−1)^𝑖11+𝜔𝑞^𝑧+𝑖⋯11+𝜔𝑞^{𝑧+𝑖−ℎ+1}𝑡^𝑛=𝑛!_∞𝑛=0[2]^ℎ_𝑞(1−𝑞)^𝑛𝑛_𝑖=0𝑛𝑖𝑞^𝑖𝑥(−1)^𝑖∞_𝑙1=0(−𝜔𝑞^𝑧+𝑖)^𝑙1⋯^∞_𝑙ℎ=0(−𝜔𝑞^{𝑧+𝑖−ℎ+1})^𝑙ℎ𝑡^𝑛=𝑛!_∞𝑛=0[2]^ℎ_𝑞(1−𝑞)^𝑛∞_{𝑙1,…,𝑙ℎ=0}(−1)^{𝑙1+⋯+𝑙ℎ𝑛}_𝑖=0𝑛𝑖𝑞^{(𝑥+𝑙1+⋯+𝑙ℎ)𝑖}(−1)^𝑖×𝜔^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1𝑧+𝑙2(𝑧−1)+⋯+𝑙ℎ(𝑧−ℎ+1)}𝑡^𝑛𝑛!=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−1)^{𝑙1+⋯+𝑙ℎ}𝜔^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1𝑧+𝑙2(𝑧−1)+⋯+𝑙ℎ(𝑧−ℎ+1)}×^∞_𝑛=0[𝑥+𝑙₁+⋯+𝑙_ℎ]^𝑛_𝑞𝑡^𝑛𝑛!=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜔)^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1𝑧+𝑙2(𝑧−1)+⋯+𝑙ℎ(𝑧−ℎ+1)}𝑒^{[𝑥+𝑙1+⋯+𝑙ℎ]𝑞𝑡}. (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_𝑞𝑒^{𝑛𝑡/(1−𝑞)}_𝑗=0(−1)^𝑗𝑞^𝑗𝑥(1)1−𝑞^𝑗11+𝜔𝑞^𝑧+𝑗⋯11+𝜔𝑞^{𝑧+𝑗−ℎ+1}=𝐸^(𝑧,ℎ)_{𝑛,𝑞,𝜔}(𝑥). (3.5)

From now on, we assume that 𝑞∈ℂ with |𝑞|<1. From (3.2) and (3.4), we note that𝐺^(𝑧,ℎ)_𝑞,𝜔(𝑥,𝑡)=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜔)^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1𝑧+𝑙2(𝑧−1)+⋯+𝑙ℎ(𝑧−ℎ+1)}𝑒^{[𝑥+𝑙1+⋯+𝑙ℎ]𝑞𝑡}=^∞_𝑛=0𝐸^(𝑧,ℎ)_{𝑛,𝑞,𝜔}𝑡(𝑥)^𝑛,𝑛!𝐸^(𝑧,ℎ)_{𝑛,𝑞,𝜔}=(𝑥)[2]^ℎ_𝑞(1−𝑞)^𝑛𝑛_𝑖=0𝑛𝑖𝑞^𝑖𝑥(−1)^𝑖11+𝜔𝑞^𝑧+𝑖⋯11+𝜔𝑞^{𝑧+𝑖−ℎ+1}=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜔)^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1𝑧+𝑙2(𝑧−1)+⋯+𝑙ℎ(𝑧−ℎ+1)}[𝑥+𝑙₁+⋯+𝑙_ℎ]^𝑛_𝑞. (3.6)(3.7)Thus we can define the multiple twisted Carlitz's type 𝑞-zeta functions as follows:𝜁^(𝑧,ℎ)_𝑞,𝜔(𝑠,𝑥)=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜔)^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1𝑧+𝑙2(𝑧−1)+⋯+𝑙ℎ(𝑧−ℎ+1)}[𝑥+𝑙₁+⋯+𝑙_ℎ]^𝑠_𝑞. (3.8)

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, then𝜁^(𝑧,ℎ)_𝑞,𝜔(𝑠,𝑥)→𝜁^(𝑧)_𝑞,𝜔(𝑠,𝑥)=[2]_𝑞^∞_𝑙=0(−𝜔)^𝑙𝑞^𝑙𝑧[𝑥+𝑙]^𝑠_𝑞. (3.9)

In [13], Wakayama and Yamasaki studied 𝑞-analogue of the Hurwitz zeta function𝜁(𝑠,𝑥)=^∞_𝑛=01(𝑛+𝑥)^𝑠 (3.10)defined by the 𝑞-series with two complex variable 𝑠,𝑧∈ℂ: 𝜁_𝑞^(𝑧)(𝑠,𝑥)=^∞_𝑛=0𝑞^{(𝑛+𝑥)𝑧}[𝑥+𝑛]^𝑠_𝑞,Re(𝑧)>0, (3.11)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:𝐸^(𝑧,ℎ)_{𝑚,𝑞,𝜔}(𝑥)=(dd𝑡)^𝑚𝐺^(𝑧,ℎ)_𝑞,𝜔(𝑥,𝑡)|_𝑡=0=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜔)^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1𝑧+𝑙2(𝑧−1)+⋯+𝑙ℎ(𝑧−ℎ+1)}[𝑥+𝑙₁+⋯+𝑙_ℎ]^𝑚_𝑞 (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 have𝜁^(𝑧,ℎ)_𝑞,𝜔(−𝑚,𝑥)=𝐸^(𝑧,ℎ)_{𝑚,𝑞,𝜔}(𝑥), (3.13)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].

4. Remarks

For nonnegative integers 𝑚 and 𝑛, we define the 𝑞-binomial coefficient [^𝑚_𝑛]_𝑞 by𝑚𝑛_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_{𝑙1,…,𝑙ℎ𝑙≥01+⋯+𝑙ℎ=𝑙}𝑞^{−(𝑙1+2𝑙2+⋯+ℎ𝑙ℎ)}=𝑞^−𝑙ℎ𝑙+ℎ−1ℎ−1_𝑞 (4.2)(cf. [12, Lamma 2.3]). From (3.8), it is easy to see that𝜁^(𝑧,ℎ)_𝑞,𝜔(𝑠,𝑥)=[2]^ℎ_𝑞^∞_{𝑙=0𝑙1,…,𝑙ℎ𝑙≥01+⋯+𝑙ℎ=𝑙}(−𝜔)^{𝑙1+⋯+𝑙ℎ}𝑞^{(𝑧+1)(𝑙1+⋯+𝑙ℎ)−(𝑙1+2𝑙2+⋯+ℎ𝑙ℎ)}[𝑥+𝑙₁+⋯+𝑙_ℎ]^𝑠_𝑞=[2]^ℎ_𝑞^∞_𝑙=0(−𝜔)^𝑙𝑞^(𝑧+1)𝑙[𝑙+𝑥]^𝑠_{𝑞𝑙1,…,𝑙ℎ𝑙≥01+⋯+𝑙ℎ=𝑙}𝑞^{−(𝑙1+2𝑙2+⋯+ℎ𝑙ℎ)}=[2]^ℎ_𝑞^∞_𝑙=0𝑙+ℎ−1ℎ−1_𝑞(−𝜔)^𝑙𝑞^{(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+ℎ−1ℎ−1_𝑞=1[ℎ−1]_𝑞!^ℎ−1_𝑗=1([𝑙+𝑥]_𝑞−𝑞^𝑙+𝑗[𝑥−𝑗]_𝑞)=^ℎ−1_𝑘=0𝑞^{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 by𝑃^𝑘_𝑞,ℎ(𝑥)=(−1)^{ℎ−1−𝑘}[ℎ−1]_𝑞!_{1≤𝑚1<⋯<𝑚ℎ−1−𝑘≤ℎ−1}𝑞^{𝑚1+⋯+𝑚ℎ−1−𝑘}[𝑥−𝑚₁]_𝑞⋯[𝑥−𝑚_{ℎ−1−𝑘}]_𝑞 (4.5)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 have𝜁^(𝑧,ℎ)_𝑞,𝜔(𝑠,𝑥)=[2]^ℎ_𝑞^ℎ−1_𝑘=0𝑃^𝑘_𝑞,ℎ(𝑥)^∞_𝑙=0(−𝜔)^𝑙𝑞^{(𝑧−𝑘)𝑙}[𝑙+𝑥]_𝑞^𝑠−𝑘=[2]_𝑞^{ℎ−1ℎ−1}_𝑘=0𝑃^𝑘_𝑞,ℎ(𝑥)𝜁^{(𝑧−𝑘)}_𝑞,𝜔(𝑠−𝑘,𝑥), (4.6)and so𝜁^(𝑧,ℎ)_𝑞,𝜔(−𝑚,𝑥)=[2]∑_𝑞^{ℎ−1ℎ−1}_𝑘=0𝑃^𝑘_𝑞,ℎ(𝑥)𝜁^{(𝑧−𝑘)}_𝑞,𝜔(−𝑚−𝑘,𝑥). (4.7)The values of 𝜁^(𝑧,ℎ)_𝑞,𝜔(−𝑚,𝑥) at ℎ=2,3 are given explicitly as follows:𝜁^(𝑧,2)_𝑞,𝜔(−𝑚,𝑥)=(1+𝑞)(𝜁^(𝑧−1)_𝑞,𝜔(−𝑚−1,𝑥)−𝑞[𝑥−1]_𝑞𝜁^(𝑧)_𝑞,𝜔𝜁(−𝑚,𝑥)),_{(𝑧,3)𝑞,𝜔}(−𝑚,𝑥)=(1+𝑞){𝜁^(𝑧−2)_𝑞,𝜔(−𝑚−2,𝑥)−(𝑞[𝑥−1]_𝑞+𝑞²[𝑥−2]_𝑞)𝜁^(𝑧−1)_𝑞,𝜔(−𝑚−1,𝑥)+𝑞³[𝑥−1]_𝑞[𝑥−2]_𝑞𝜁^(𝑧)_𝑞,𝜔(−𝑚,𝑥)}. (4.8)

Acknowledgment

This work is supported by Kyungnam University Foundation Grant no. 2007.