On Multiple Twisted 𝑝-adic 𝑞-Euler 𝜁-Functions and 𝑙-Functions

Abstract

We give the existence of multiple twisted 𝑝-adic 𝑞-Euler 𝜁-functions and 𝑙-functions, which are generalization of the twisted 𝑝-adic (ℎ,𝑞)-zeta functions and twisted 𝑝-adic (ℎ,𝑞)-Euler 𝑙-functions in the work of Ozden and Simsek (2008).

1. Introduction, Definitions, and Notations

The constants 𝐸_𝑛 in the Taylor series expansion2𝑒^𝑡=+1^∞_𝑛=0𝐸_𝑛𝑡^𝑛𝑛!(1.1)are known as the first kind Euler numbers (cf. [1]). From the generating function of the first kind Euler numbers, we note that 𝐸₀=1 and 𝐸_𝑛=−∑^𝑛_𝑘=0(^𝑛_𝑘)𝐸_𝑘 for 𝑛∈ℕ. The first few are 1,−1/2,0,1/4,−1/2,… and 𝐸_2𝑘=0 for 𝑘∈ℕ. Those numbers play an important role in number theory. For example, the Euler zeta-function essentially equals an Euler numbers at nonpositive integer:𝜁_𝐸(−𝑚)=𝐸_𝑚for𝑚≥0,(1.2)where𝜁_𝐸(𝑠)=^∞_𝑛=1(−1)^𝑛𝑛^𝑠,𝑠∈ℂ(1.3)(see [1–10]).

Throughout this paper ℤ,ℤ_𝑝,ℚ_𝑝, and ℂ_𝑝 will denote the ring of integers, the ring of 𝑝-adic rational integers, the field of 𝑝-adic rational numbers, and the completion of the algebraic closure of ℚ_𝑝, respectively. Let 𝑣_𝑝 be the normalized exponential valuation of ℂ_𝑝 with |𝑝|_𝑝=1/𝑝. 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<1, so that 𝑞^𝑥=exp(𝑥log𝑞) for 𝑥∈ℤ_𝑝. If 𝑞∈ℂ, then we assume that |𝑞|<1. Also we use the following notations:[𝑥]_𝑞=1−𝑞^𝑥1−𝑞,[𝑥]_−𝑞=1−(−𝑞)^𝑥,1+𝑞(1.4)cf. [2–4]. For𝑓∈𝑈𝐷(ℤ_𝑝,ℂ_𝑝)={𝑓∣𝑓∶ℤ_𝑝→ℂ_𝑝isuniformlydifferentiablefunction},(1.5)the 𝑝-adic 𝑞-integral on ℤ_𝑝 was defined by Kim (cf. [2–4]) as follows:𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝑎)𝑑𝜇_𝑞(𝑎)=lim_𝑁→∞1[𝑑𝑝^𝑁]_𝑞^{𝑑𝑝𝑁−1}_𝑎=0𝑓(𝑎)𝑞^𝑎for|1−𝑞|_𝑝<1.(1.6)Furthermore, we can consider the fermionic integral in contrast to the conventional bosonic integral. That is, 𝐼₋₁(𝑓)=∫_ℤ𝑝𝑓(𝑎)𝑑𝜇₋₁(𝑎) (cf. [5]). From this, we derive𝐼₋₁(𝑓₁)+𝐼₋₁(𝑓)=2𝑓(0),(1.7)where 𝑓₁(𝑎)=𝑓(𝑎+1). Substitute 𝑓(𝑎)=𝜉^𝑎𝑞^𝛼𝑎𝑒^𝑎𝑡 into (1.7). The twisted (𝛼,𝑞)-extension of Euler numbers is defined by [8]𝐼₋₁(𝜉^𝑎𝑞^𝛼𝑎𝑒^𝑎𝑡2)=𝜉𝑞^𝛼𝑒^𝑡=+1^∞_𝑛=0𝐸^(𝛼)_𝑛,𝜉𝑡(𝑞)^𝑛.𝑛!(1.8)For |1−𝑞|_𝑝<1, we consider fermionic 𝑝-adic 𝑞-integral on ℤ_𝑝 which is the 𝑞-extension of 𝐼₋₁(𝑓) as follows:𝐼_−𝑞(𝑓)=_ℤ𝑝𝑓(𝑎)𝑑𝜇_−𝑞(𝑎)=lim_𝑁→∞1[𝑑𝑝^𝑁]_−𝑞^{𝑑𝑝𝑁−1}_𝑎=0𝑓(𝑎)(−𝑞)^𝑎(1.9)(cf. [5]). From (1.9), we can derive the following formula [5]:𝑞𝐼_−𝑞(𝑓₁)+𝐼_−𝑞(𝑓)=[2]_𝑞𝑓(0),(1.10)where 𝑓₁(𝑎) is translation with 𝑓₁(𝑎)=𝑓(𝑎+1). If we take 𝑓(𝑎)=𝜉^𝑎𝑒^𝑎𝑡, then we have 𝑓₁(𝑎)=𝜉^𝑎+1𝑒^(𝑎+1)𝑡=𝜉^𝑎𝑒^𝑎𝑡𝜉𝑒^𝑡. From (1.10), we derive (𝜉𝑞𝑒^𝑡+1)𝐼_−𝑞(𝜉^𝑎𝑒^𝑎𝑡)=[2]_𝑞. Hence, we obtain𝐼_−𝑞(𝜉^𝑎𝑒^𝑎𝑡)=_ℤ𝑝𝜉^𝑎𝑒^𝑎𝑡𝑑𝜇_−𝑞(𝑎)=[2]_𝑞𝜉𝑞𝑒^𝑡.+1(1.11)By (1.11), we define the twisted 𝑞-Euler numbers, 𝐸_{𝑛,𝑞,𝜉} by means of the following generating function (cf. [5]):[2]_𝑞𝜉𝑞𝑒^𝑡=+1^∞_𝑛=0𝐸_{𝑛,𝑞,𝜉}𝑡^𝑛.𝑛!(1.12)These numbers are interpolated by the twisted Euler 𝑞-zeta function which is defined as follows:𝜁_{𝑞,𝜉,𝐸}(𝑠)=[2]_𝑞^∞_𝑛=1(−1)^𝑛𝜉^𝑛𝑞^𝑛𝑛^𝑠,𝑠∈ℂ.(1.13)Note that 𝜁_{𝑞,𝜉,𝐸}(𝑠) is analytic function in the whole complex plane ℂ.

In view of the functional equation for the twisted Euler 𝑞-zeta function at nonpositive integers, we have𝜁_{𝑞,𝜉,𝐸}(−𝑚)=𝐸_{𝑚,𝑞,𝜉}for𝑚≥0(1.14)(cf. [5]).

Twisted 𝑞-Bernoulli and Euler numbers and polynomials are very important not only in practically every field of mathematics, in particular in combinatorial theory, finite difference calculus, numerical analysis, numbers theory, but also probability theory. Recently the 𝑞-extensions of those Euler numbers (polynomials) and the multiple of 𝑞-extensions of those Euler numbers (polynomials) have been studied by many authors, (cf. [1–15]). In [8], Ozden and Simsek have studied (ℎ,𝑞)-extensions of twisted Euler numbers and polynomials by using 𝑝-adic 𝑞-integral on the ring of 𝑝-adic integers ℤ_𝑝. From their (ℎ,𝑞)-extensions of twisted Euler numbers and polynomials, they have derived 𝑝-adic (ℎ,𝑞)-extensions of Euler zeta function and 𝑝-adic (ℎ,𝑞)-extensions of Euler 𝑙-functions. They also gave some interesting relations between their (ℎ,𝑞)-Euler numbers and (ℎ,𝑞)-Euler zeta functions, and found the 𝑝-adic twisted interpolation function of the generalized twisted (ℎ,𝑞)-Euler numbers. In [11], Jang defined twisted 𝑞-Euler numbers and polynomials of higher order, and studied multiple twisted 𝑞-Euler zeta functions. He also derived sums of products of 𝑞-Euler numbers and polynomials by using fermionic 𝑝-adic 𝑞-integral. In [7, 9], 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 [16], Kim constructed multiple 𝑝-adic 𝐿-functions, which interpolate the Bernoulli numbers of higher order. He also derived that the values of the partial derivative of this multiple 𝑝-adic 𝐿-function at 𝑠=0 are given.

In this paper, we consider twisted 𝑞-Euler numbers and polynomials of higher order, and study multiple twisted 𝑝-adic, 𝑞-Euler, 𝜁-functions, and 𝑙-functions, which are generalization of the twisted 𝑝-adic (ℎ,𝑞)-zeta functions and twisted 𝑝-adic (ℎ,𝑞)-Euler 𝑙-functions in [8].

2. Preliminaries

We assume that 𝑞∈ℂ with |𝑞|<1. Let 𝜉 be a primitive 𝑟 th root of unity.

For an integer ℎ, the twisted 𝑞-Euler polynomials of higher order (the index ℎ may be negative), 𝐸^(ℎ)_{𝑛,𝑞,𝜉}(𝑥), are defined by means of the following generating function (cf. [11, 14]):𝐹^(ℎ)_𝑞,𝜉(𝑡,𝑥)=[2]_𝑞1+𝜉𝑞𝑒^𝑡⋯[2]_𝑞1+𝜉𝑞𝑒^𝑡_ℎ-times𝑒^𝑥𝑡=[2]^ℎ_𝑞𝑒^∞𝑡𝑥_𝑙1=0(−𝜉)^𝑙1𝑞^𝑙1𝑒^𝑙1𝑡⋯^∞_𝑙ℎ=0(−𝜉)^𝑙ℎ𝑞^𝑙ℎ𝑒^𝑙ℎ𝑡=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜉𝑞)^{𝑙1+⋯+𝑙ℎ}𝑒^{(𝑙1+⋯+𝑙ℎ+𝑥)𝑡}=^∞_𝑛=0𝐸^(ℎ)_{𝑛,𝑞,𝜉}𝑡(𝑥)^𝑛,𝑛!(2.1)where |𝑡+log(𝜉𝑞)|<𝜋. Note that [2]_𝑞=1+𝑞, so [2]_𝑞/(1+𝜉𝑞𝑒^𝑡)≡(1+𝑞)/(1+𝜉𝑞)(mod𝑡). Of course the explicit formulas in (2.1) depend on ℎ which is a positive integer. If ℎ=1,𝑞=𝑞^𝛼 in the above, we obtain the generating function of the twisted (𝛼,𝑞)-extension of Euler polynomials in [8, cf. Section 1, (1.3)]. In fact, if ℎ>0 then −ℎ<0. Therefore, the generating function 𝐹^(−ℎ)_𝑞,𝜉(𝑡,𝑥) is the form𝐹^(−ℎ)_𝑞,𝜉(𝑡,𝑥)=([2]_𝑞1+𝜉𝑞𝑒^𝑡)^−ℎ𝑒^𝑡𝑥=(1+𝜉𝑞𝑒^𝑡[2]_𝑞)^ℎ𝑒^𝑡𝑥=^∞_𝑛=0𝐸^(−ℎ)_{𝑛,𝑞,𝜉}𝑡(𝑥)^𝑛.𝑛!(2.2)The twisted 𝑞-Euler numbers of higher order are 𝐸^(ℎ)_{𝑛,𝑞,𝜉}=𝐸^(ℎ)_{𝑛,𝑞,𝜉}(0). Then, it is immediate that𝐸^(ℎ)_{𝑛,𝑞,𝜉}(𝑥)=^𝑛_𝑘=0𝑛𝑘𝐸^(ℎ)_{𝑘,𝑞,𝜉}𝑥^𝑛−𝑘.(2.3)

From now on, we assume ℎ>0 and in general whenever ℎ is actually an index then ℎ>0. Jang [11] defined the two-variable multiple twisted 𝑞-Euler zeta functions as follows. Definition 2.1. For 𝑠∈ℂ and 𝑥∈ℝ⁺={𝑥∈ℝ∣𝑥>0}, one defines𝜁^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥)=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜉)^{𝑙1+⋯+𝑙ℎ}𝑞^{𝑙1+⋯+𝑙ℎ}(𝑙₁+⋯+𝑙_ℎ+𝑥)^𝑠.(2.4)

𝜁^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥) is an analytical function in the whole complex plane.

The value of 𝜁^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥) at nonpositive integers, ℤ₊=ℕ∪{0}, is given explicitly as follows. Theorem 2.2 (see [11]). Let 𝑚∈ℤ₊. Then, 𝜁^(ℎ)_{𝑞,𝜉,𝐸}(−𝑚,𝑥)=𝐸^(ℎ)_{𝑚,𝑞,𝜉}(𝑥).

Let 𝜒 be a Dirichlet character with odd conductor 𝑑. We define a twisted Dirichlet's type 𝑞-Euler polynomials of higher order by means of the following generating function (cf. [11, 14]):𝐹^(ℎ)_{𝑞,𝜉,𝜒}=1(𝑡,𝑥)[𝑑]^ℎ_−𝑞^𝑑−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝑒^{(𝑎1+⋯+𝑎ℎ)𝑡}1+𝑞^𝑑1+𝜉^𝑑𝑞^𝑑𝑒^d𝑡⋯1+𝑞^𝑑1+𝜉^𝑑𝑞^𝑑𝑒^d𝑡_ℎ-times𝑒^𝑥𝑡=[2]^ℎ_𝑞^𝑑−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝑒^{(𝑎1+⋯+𝑎ℎ∞)𝑡}_𝑥1=0(−𝜉^𝑑𝑞^𝑑𝑒^d𝑡)^𝑥1⋯^∞_𝑥ℎ=0(−𝜉^𝑑𝑞^𝑑𝑒^d𝑡)^𝑥ℎ_ℎ-times𝑒^𝑥𝑡=[2]^ℎ_𝑞^∞_{𝑥1,…,𝑥ℎ=0}^𝑑−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+𝑑𝑥₁+⋯+𝑎_ℎ+𝑑𝑥_ℎ)(−𝜉𝑞)^{𝑎1+𝑑𝑥1+⋯+𝑎ℎ+𝑑𝑥ℎ}𝑒^{(𝑎1+𝑑𝑥1+⋯+𝑎ℎ+𝑑𝑥ℎ)𝑡}𝑒^𝑥𝑡=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}𝜒(𝑙₁+⋯+𝑙_ℎ)(−𝜉𝑞)^{𝑙1+⋯+𝑙ℎ}𝑒^{(𝑥+𝑙1+⋯+𝑙ℎ)𝑡}=^∞_𝑛=0𝐸^(ℎ)_{𝑛,𝑞,𝜉,𝜒}𝑡(𝑥)^𝑛.𝑛!(2.5)

We now see that the twisted Dirichlet's type 𝑞-Euler polynomials of higher order are easily expressed by the twisted 𝑞-Euler polynomials of higher order as follows. Proposition 2.3. Let 𝐹 be an odd multiple of the conductor 𝑑. Then,𝐸^(ℎ)_{𝑛,𝑞,𝜉,𝜒}(𝑥)=𝐹^𝑛1[𝐹]^ℎ_−𝑞^𝐹−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝐸^(ℎ)_{𝑛,𝑞𝐹,𝜉𝐹}(𝑎₁+⋯+𝑎_ℎ+𝑥𝐹).(2.6)Proof. Let 𝑑(=odd)∈ℕ. By (2.1) and (2.5), we note that𝐹^(ℎ)_{𝑞,𝜉,𝜒}1(𝑡,𝑥)=[𝑑]^ℎ_−𝑞^𝑑−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝐹_𝑞^(ℎ)_{𝑑,𝜉𝑑}(d𝑎𝑡,¹+⋯+𝑎_ℎ+𝑥𝑑).(2.7)Then, we have𝐸^(ℎ)_{𝑛,𝑞,𝜉,𝜒}(𝑥)=𝑑^𝑛1[𝑑]^ℎ_−𝑞^𝑑−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝐸^(ℎ)_{𝑛,𝑞𝑑,𝜉𝑑}(𝑎₁+⋯+𝑎_ℎ+𝑥𝑑).(2.8)On the other hand, if 𝐹=𝑑𝑝, then we get1[𝐹]^ℎ_−𝑞^𝐹−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝐹_𝑞^(ℎ)_{𝐹,𝜉𝐹}𝑎(𝐹𝑡,¹+⋯+𝑎_ℎ+𝑥𝐹)=^𝐹−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}([2]_𝑞𝜉^𝐹𝑞^𝐹𝑒^𝐹𝑡)+1^ℎ𝑒^{(𝑎1+⋯+𝑎ℎ+𝑥)𝑡}=^𝑑−1_{𝑏1,…,𝑏ℎ=0}^𝑝−1_{𝑐1,…,𝑐ℎ=0}𝜒(𝑏₁+𝑐₁𝑑+⋯+𝑏_ℎ+𝑐_ℎ𝑑)(−𝜉𝑞)^{𝑏1+𝑐1𝑑+⋯+𝑏ℎ+𝑐ℎ𝑑}×([2]_𝑞𝜉^𝐹𝑞^𝐹𝑒^𝐹𝑡)+1^ℎ𝑒^{(𝑏1+𝑐1𝑑+⋯+𝑏ℎ+𝑐ℎ𝑑+𝑥)𝑡}=1[𝑑]^ℎ_−𝑞^𝑑−1_{𝑏1,…,𝑏ℎ=0}𝜒(𝑏₁+⋯+𝑏_ℎ)(−𝜉𝑞)^{𝑏1+⋯+𝑏ℎ}𝐹_𝑞^(ℎ)_{𝑑,𝜉𝑑}(d𝑏𝑡,¹+⋯+𝑏_ℎ+𝑥𝑑).(2.9)This completes the proof.

The two-variable multiple twisted 𝑞-Euler 𝑙-functions are defined by the following definition. Definition 2.4 (see [14]). Let 𝜒 be a Dirichlet character. For 𝑠∈ℂ and 𝑥∈ℝ⁺, one has𝑙^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥,𝜒)=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}𝜒(𝑙₁+⋯+𝑙_ℎ)∏^ℎ_𝑖=1(−1)^𝑙𝑖𝜉^𝑙𝑖𝑞^𝑙𝑖(𝑙₁+⋯+𝑙_ℎ+𝑥)^𝑠.(2.10)

The value of 𝑙^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥,𝜒) at nonpositive integers is given explicitly by the following theorem. Theorem 2.5 (see [14]). Let 𝑚∈ℤ₊. Then 𝑙^(ℎ)_{𝑞,𝜉,𝐸}(−𝑚,𝑥,𝜒)=𝐸^(ℎ)_{𝑚,𝑞,𝜉,𝜒}(𝑥).Proof. Let 𝜒 be a Dirichlet character with odd conductor 𝑑 and let 𝐹 be an odd number of multiple 𝑑. Set 𝑠∈ℂ and 𝑥∈ℝ⁺. Beside the multiple twisted 𝑞-Euler 𝑙-function 𝑙^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥,𝜒), we consider the multiple twisted 𝑞-Euler zeta function 𝜁^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥) in Definition 2.1. Then𝑙^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥,𝜒)=𝐹^−𝑠1[𝐹]^ℎ_−𝑞^𝐹−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝜁_𝑞^(ℎ)_{𝐹,𝜉𝐹,𝐸}𝑎(𝑠,¹+⋯+𝑎_ℎ+𝑥𝐹)(2.11)(cf. [14]).
In the integral for Γ(𝑠), we make the change of variable 𝑦=(𝑥+𝑙₁+⋯+𝑙_ℎ)𝑡, where 𝑙₁,…,𝑙_ℎ≥0, to obtainΓ(𝑠)=^∞₀𝑒^−𝑦𝑦^𝑠d𝑦𝑦=(𝑥+𝑙₁+⋯+𝑙_ℎ)^𝑠∞₀𝑒^{−(𝑥+𝑙1+⋯+𝑙ℎ)𝑡}𝑡^𝑠d𝑡𝑡,(2.12)or(𝑥+𝑙₁+⋯+𝑙_ℎ)^−𝑠Γ(𝑠)=∫^∞₀𝑒^{−(𝑥+𝑙1+⋯+𝑙ℎ)𝑡}𝑡^𝑠d𝑡𝑡.(2.13)Summing over all 𝑙₁,…,𝑙_ℎ≥0, we find[2]∑^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜉𝑞)^{𝑙1+⋯+𝑙ℎ}(𝑥+𝑙₁+⋯+𝑙_ℎ)^−𝑠Γ(𝑠)=[2]∑^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜉𝑞)^{𝑙1+⋯+𝑙ℎ}∫^∞₀𝑒^{−(𝑥+𝑙1+⋯+𝑙ℎ)𝑡}𝑡^𝑠d𝑡𝑡.(2.14)This gives us𝜁^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥)Γ(𝑠)=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}(−𝜉𝑞)^{𝑙1+⋯+𝑙ℎ∞}₀𝑒^{−(𝑥+𝑙1+⋯+𝑙ℎ)𝑡}𝑡^𝑠d𝑡𝑡=^∞₀𝑡^𝑠[2]_𝑞1+𝜉𝑞𝑒^−𝑡⋯[2]_𝑞1+𝜉𝑞𝑒^−𝑡_ℎ-times𝑒^−𝑥𝑡d𝑡𝑡=^∞₀𝑡^𝑠𝐹^(ℎ)_𝑞,𝜉(−𝑡,𝑥)d𝑡𝑡.(2.15)If we divide the infinite integral into two parts:^∞₀𝑡^𝑠𝐹^(ℎ)_𝑞,𝜉(−𝑡,𝑥)d𝑡𝑡=¹₀𝑡^𝑠𝐹^(ℎ)_𝑞,𝜉(−𝑡,𝑥)d𝑡𝑡+^∞₁𝑡^𝑠𝐹^(ℎ)_𝑞,𝜉(−𝑡,𝑥)d𝑡𝑡,(2.16)it is easily seen that the second term is an entire function on 𝑡.
Consider ∫¹₀𝑡^𝑠𝐹^(ℎ)_𝑞,𝜉(−𝑡,𝑥)(d𝑡/𝑡). By the definition of 𝐸^(ℎ)_{𝑛,𝑞,𝜉}(𝑥), we have𝐹^(ℎ)_𝑞,𝜉(−𝑡,𝑥)=^∞_𝑛=0𝐸^(ℎ)_{𝑛,𝑞,𝜉}(𝑥)(−1)^𝑛𝑡^𝑛𝑛!=([2]_𝑞1+𝜉𝑞𝑒^−𝑡)^ℎ𝑒^−𝑥𝑡.(2.17)Therefore,¹₀𝑡^𝑠𝐹^(ℎ)_𝑞,𝜉(−𝑡,𝑥)d𝑡𝑡=^∞_𝑛=0𝐸^(ℎ)_{𝑛,𝑞,𝜉}(𝑥)𝑛!(−1)^𝑛1₀𝑡^{𝑠+𝑛−1}d𝑡=^∞_𝑛=0𝐸^(ℎ)_{𝑛,𝑞,𝜉}(𝑥)𝑛!(−1)^𝑛.𝑠+𝑛(2.18)This has an analytic continuation to a meromorphic function 𝑠 in the entire complex plane. It is holomorphic except at 𝑠=0,−1,−2,…, where it has a pole of order 1. Note that Γ(𝑠) is holomorphic except at 𝑠=0,−1,−2,…, where it has a pole of order 1. Γ(𝑠) does not have a zero. Therefore, 𝜁^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥) has an analytic continuation to the whole complex plane. For an integer 𝑚∈ℤ₊, we havelim_{𝑠→−𝑚}(𝑠+𝑚)(𝜁^(ℎ)_{𝑞,𝜉,𝐸}𝐸(𝑠,𝑥)Γ(𝑠))=^{(ℎ)𝑚,𝑞,𝜉}(𝑥)𝑚!(−1)^𝑚.(2.19)If 𝑚∈ℤ₊, we have lim_{𝑠→−𝑚}(𝑠+𝑚)Γ(𝑠)=(−1)^𝑚(1/𝑚!), and thus we obtain𝜁^(ℎ)_{𝑞,𝜉,𝐸}(−𝑚,𝑥)=𝐸^(ℎ)_{𝑚,𝑞,𝜉}(𝑥).(2.20)Consequently, by using Propositions 2.3 and (2.6) and the above equation, we have𝑙^(ℎ)_{𝑞,𝜉,𝐸}(−𝑚,𝑥,𝜒)=𝐹^𝑚1[𝐹]^ℎ_−𝑞^𝐹−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝐸^(ℎ)_{𝑚,𝑞𝐹,𝜉𝐹}(𝑎₁+⋯+𝑎_ℎ+𝑥𝐹)=𝐸^(ℎ)_{𝑚,𝑞,𝜉,𝜒}(𝑥).(2.21)Therefore, we obtain another proof of Theorem 2.5.Remark 2.6 (see [11, 14]). We put𝐷=(dd𝑡).(2.22)Let 𝑚∈ℤ₊, and let 𝑥∈ℝ⁺. From (2.1) and (2.4), we obtain the following:𝐸^(ℎ)_{𝑚,𝑞,𝜉}(𝑥)=𝐷^𝑚𝐹^(ℎ)_𝑞,𝜉(𝑡,𝑥)|_𝑡=0=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙𝑘=0}(−𝜉𝑞)^{𝑙1+⋯+𝑙ℎ}(𝑥+𝑙₁+⋯+𝑙_ℎ)^𝑚=𝜁^(ℎ)_{𝑞,𝜉,𝐸}(−𝑚,𝑥).(2.23)Similarly, by (2.5) and (2.11), we have𝐸^(ℎ)_{𝑚,𝑞,𝜉,𝜒}(𝑥)=𝐷^𝑚𝐹^(ℎ)_{𝑞,𝜉,𝜒}(𝑡,𝑥)|_𝑡=0=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ=0}𝜒(𝑙₁+⋯+𝑙_ℎ)^ℎ_𝑖=1(−1)^𝑙𝑖𝜉^𝑙𝑖𝑞^𝑙𝑖(𝑥+𝑙₁+⋯+𝑙_ℎ)^𝑚=𝑙^(ℎ)_{𝑞,𝜉,𝐸}(−𝑚,𝑥,𝜒).(2.24)

3. Partial Multiple Twisted 𝑞-Euler 𝜁-Functions

Let 𝑠∈ℂ and 𝑎_𝑖,𝐹∈ℤ with 𝐹 as an odd integer and 0<𝑎_𝑖<𝐹, where 𝑖=1,…,ℎ. Then, partial multiple twisted 𝑞-Euler 𝜁-functions are as follows (cf. [14, 16, 18–20]):𝐻^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑎₁,…,𝑎_ℎ,𝑥∣𝐹)=[2]^ℎ_𝑞^∞_{𝑙1,…,𝑙ℎ𝑙=0𝑖≡𝑎𝑖(mod𝐹),𝑖=1,…,ℎ}(−𝜉𝑞)^{𝑙1+⋯+𝑙ℎ}(𝑙₁+⋯+𝑙_ℎ+𝑥)^𝑠.(3.1)We give a relationship between 𝐻^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑎₁,…,𝑎_ℎ,𝑥∣𝐹) and 𝜁_𝑞^(ℎ)_{𝐹,𝜉𝐹,𝐸}(𝑠,𝑥) as follows. For 𝑖=1,…,ℎ, substituting 𝑙_𝑖=𝑎_𝑖+𝑛_𝑖𝐹 with 𝐹 as an odd into (3.1), we have𝐻^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑎₁,…,𝑎_ℎ,𝑥∣𝐹)=[2]^ℎ_𝑞^∞_{𝑛1,…,𝑛ℎ=0}(−𝜉𝑞)^{𝑎1+𝑛1𝐹+⋯+𝑎ℎ+𝑛ℎ𝐹}(𝑎₁+𝑛₁𝐹+⋯+𝑎_ℎ+𝑛_ℎ𝐹+𝑥)^𝑠=1[𝐹]^ℎ_−𝑞(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝐹^𝑠[2]^ℎ_𝑞𝐹^∞_{𝑛1,…,𝑛ℎ=0}(−𝜉^𝐹𝑞^𝐹)^{𝑛1+⋯+𝑛ℎ}(𝑛₁+⋯+𝑛_ℎ+(𝑎₁+⋯+𝑎_ℎ+𝑥)/𝐹)^𝑠=1[𝐹]^ℎ_−𝑞(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝐹^𝑠𝜁_𝑞^(ℎ)_{𝐹,𝜉𝐹,𝐸}𝑎(𝑠,¹+⋯+𝑎_ℎ+𝑥𝐹).(3.2)

By using (2.3) and Theorem 2.2 and substituting 𝑠=−𝑚,𝑚∈ℤ₊ in the above, we arrive at the following theorem.

Theorem 3.1. Let 𝐹 be an odd integer, 𝑠∈ℂ and let 𝑥∈ℝ⁺. Then𝐻^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑎₁,…,𝑎_ℎ1,𝑥∣𝐹)=[𝐹]^ℎ_−𝑞(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝐹^𝑠𝜁_𝑞^(ℎ)_{𝐹,𝜉𝐹,𝐸}𝑎(𝑠,¹+⋯+𝑎_ℎ+𝑥𝐹).(3.3)In particular, if 𝑚∈ℤ₊, then𝐻^(ℎ)_{𝑞,𝜉,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ=1,𝑥∣𝐹)[𝐹]^ℎ_−𝑞(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}(𝑎₁+⋯+𝑎_ℎ+𝑥)^𝑚𝑚_𝑘=0𝑚𝑘(𝐹𝑎₁+⋯+𝑎_ℎ)+𝑥^𝑘𝐸^(ℎ)_{𝑘,𝑞𝐹,𝜉𝐹}.(3.4)

By using Theorem 3.1 and (2.11), we arrive at the following theorem.

Theorem 3.2. Let 𝜒 be a Dirichlet character with conductor 𝑑 and 𝐹 as an odd multiple of 𝑑. Then,𝑙^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥,𝜒)=^𝐹−1_{𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)𝐻^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥,𝑎₁,…,𝑎_ℎ,𝑥∣𝐹),(3.5)where 𝑠∈ℂ and 𝑥∈ℝ⁺.

4. Multiple Twisted 𝑝-Adic 𝑞-Euler 𝑙-Functions

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 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^𝑁-th roots of unity in ℂ^×_𝑝=ℂ_𝑝⧵{0}. Set𝐓_𝑝={𝜉∈ℂ_𝑝∣𝜉^𝑝𝑁=1,forsome𝑁≥0}=_𝑁≥0𝐶_𝑝𝑁.(4.1)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<1.

We will consider the 𝑝-adic analogue of the 𝑙^(ℎ)_{𝑞,𝜉,𝐸}-functions which are introduced in the previous section. In order to consider 𝑝-adic and complex 𝑙^(ℎ)_{𝑞,𝜉,𝐸}-functions simultaneously, we will use an isomorphism, 𝜎, between the algebraic closure of the rational numbers in ℂ_𝑝 and the algebraic closure of the rational numbers within the complex numbers ℂ. Our purpose is to discuss the values of 𝑙^(ℎ)_{𝑞,𝜉,𝐸}-functions, so we will consider 𝜎 as fixed throughout this section and use 𝜎 to identify 𝑝-adic algebraic numbers with complex algebraic numbers. We will write 𝑥=𝑦, when 𝑥∈ℂ_𝑝,𝑦∈ℂ and 𝑦=𝜎(𝑥).

Let 𝜔 be denoted as the Teichmüller character having conductor 𝑝. For an arbitrary character 𝜒, let 𝜒_𝑛=𝜒𝜔^−𝑛, where 𝑛∈ℤ, in sense of the product of characters. We put⟨𝑎⟩=𝜔⁻¹𝑎(𝑎)𝑎=,𝜔(𝑎)(4.2)whenever (𝑎,𝑝)=1. We then have ⟨𝑎⟩≡1(mod𝑝ℤ_𝑝) for these values of 𝑎. Note that we extend this notation by defining⟨𝑎+𝑝𝑡⟩=𝜔⁻¹(𝑎)(𝑎+𝑝𝑡)(4.3)for all 𝑎∈ℤ with (𝑎,𝑝)=1, and 𝑡∈ℂ_𝑝 such that |𝑡|_𝑝≤1. Thus, ⟨𝑎+𝑝𝑡⟩=⟨𝑎⟩+𝑝𝜔⁻¹(𝑎)𝑡, so that ⟨𝑎+𝑝𝑡⟩≡1(mod𝑝ℤ_𝑝[𝑡]) (cf. [21, 22]).

The significance of Theorem 3.1 lies in the fact that the right-hand side is essentially a liner combination of terms of the form^𝑚_𝑘=0𝑚𝑘(𝐹𝑎₁+⋯+𝑎_ℎ)+𝑝𝑡^𝑘𝐸^(ℎ)_{𝑘,𝑞𝐹,𝜉𝐹},(4.4)which makes sense when 𝑚 is replaced by a 𝑝-adic variable and 𝑝|𝐹. Set𝐷={𝑠∈ℂ_𝑝∣|𝑠|_𝑝≤𝑝^{(𝑝−2)/(𝑝−1)}},(4.5)(cf. [8, 16, 18–22]). Let 𝐹 be an odd integer, and let (𝑎₁+⋯+𝑎_ℎ,𝑝)=1,𝑎_𝑖∈ℤ with 0<𝑎_𝑖<𝐹 for 𝑖=1,…,ℎ. Suppose that 𝑠∈𝐷 and 𝑡∈ℂ_𝑝 with |𝑡|_𝑝≤1. We apply [18, Proposition 5.8, page 53] to the series^∞_𝑘=0𝑠𝑘(𝐹𝑎₁+⋯+𝑎_ℎ)+𝑝𝑡^𝑘𝐸^(ℎ)_{𝑘,𝑞𝐹,𝜉𝐹}.(4.6)Let 𝑞∈1+𝐌_𝑝 where 𝐌_𝑝={𝑧∈ℂ_𝑝∣|𝑧|<1}. In [13], we see that|𝐸^(ℎ)_{𝑘,𝑞𝐹,𝜉𝐹}|_𝑝≤1,(4.7)since 𝜉∈𝐓_𝑝. Observe that we have for odd 𝑝|𝐹,𝐹|(𝑎₁+⋯+𝑎_ℎ)+𝑝𝑡^𝑘𝐸^(ℎ)_{𝑘,𝑞𝐹,𝜉𝐹}|_𝑝≤𝑝^−𝑘1=(𝑝)^𝑘,(4.8)so that we can take 𝑟=1/𝑝 and 𝑀=1 in [18, Proposition 5.8]. This prove that (4.6) is analytic in 𝐷. Note that ⟨𝑎₁+⋯+𝑎_ℎ+𝑝𝑡⟩^−𝑠 is analytic in 𝐷 for (𝑎₁+⋯+𝑎_ℎ,𝑝)=1 and 𝑡∈ℂ_𝑝 such that |𝑡|_𝑝≤1.Definition 4.1. Let 𝐹 be an odd integer, and let (𝑎₁+⋯+𝑎_ℎ,𝑝)=1,𝑎_𝑖∈ℤ with 0<𝑎_𝑖<𝐹 for 𝑖=1,…,ℎ. Suppose that 𝑠∈𝐷 and 𝑡∈ℂ_𝑝 with |𝑡|_𝑝≤1. One defines the partial multiple twisted 𝑝-adic 𝑞-Euler 𝜁-functions for 𝑝∣𝐹: 𝐻^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(𝑠,𝑎₁,…,𝑎_ℎ=,𝑝𝑡∣𝐹)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}[𝐹]^ℎ_−𝑞⟨𝑎₁+⋯+𝑎_ℎ+𝑝𝑡⟩^∞−𝑠_𝑘=0𝑘(−𝑠𝐹𝑎₁+⋯+𝑎_ℎ)+𝑝𝑡^𝑘𝐸^(ℎ)_{𝑘,𝑞𝐹,𝜉𝐹}.(4.9)

Theorem 4.2. Let 𝐹 be an odd integer with 𝑝∣𝐹, and let (𝑎₁+⋯+𝑎_ℎ,𝑝)=1,𝑎_𝑖∈ℤ with 0<𝑎_𝑖<𝐹 for 𝑖=1,…,ℎ. Suppose that 𝑠∈𝐷 and 𝑡∈ℂ_𝑝 with |𝑡|_𝑝≤1. Then 𝐻^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(𝑠,𝑎₁,…,𝑎_ℎ,𝑝𝑡∣𝐹) is a 𝑝-adic analytic function on 𝐷 such that𝐻^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ,𝑝𝑡∣𝐹)=𝜔^−𝑚(𝑎₁+⋯+𝑎_ℎ)𝐻^(ℎ)_{𝑞,𝜉,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ,𝑝𝑡∣𝐹)(4.10)for 𝑚∈ℤ₊. In particular, if 𝑚≡0(mod𝑝−1), then𝐻^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ,𝑝𝑡∣𝐹)=𝐻^(ℎ)_{𝑞,𝜉,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ,𝑝𝑡∣𝐹).(4.11)Proof. We have already remarked that ⟨𝑎₁+⋯+𝑎_ℎ+𝑝𝑡⟩^−𝑠 is analytic in 𝐷 for (𝑎₁+⋯+𝑎_ℎ,𝑝)=1 and 𝑡∈ℂ_𝑝 such that |𝑡|_𝑝≤1. Also we see that (4.6) is analytic in 𝐷. It is clear that 𝐻^(ℎ)_{𝑝,𝑞,𝜉,𝐸} is a 𝑝-adic analytic function on 𝐷. For 𝑠=−𝑚,𝑚∈ℤ₊ one has𝐻^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ=,𝑝𝑡∣𝐹)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}[𝐹]^ℎ_−𝑞(𝑎₁+⋯+𝑎_ℎ+𝑝𝑡𝜔(𝑎₁+⋯+𝑎_ℎ))^𝑚𝑚_𝑘=0𝑚𝑘(𝐹𝑎₁+⋯+𝑎_ℎ)+𝑝𝑡^𝑘𝐸^(ℎ)_{𝑘,𝑞𝐹,𝜉𝐹}(whereweuseTheorem3.1)=𝜔^−𝑚(𝑎₁+⋯+𝑎_ℎ)𝐻^(ℎ)_{𝑞,𝜉,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ,𝑝𝑡∣𝐹).(4.12)This completes the proof.Definition 4.3. Let 𝜒 be a Dirichlet character with odd conductor 𝑑, and let 𝐹 be a positive multiple of 𝑝 and 𝑑. We can define the multiple twisted 𝑝-adic 𝑞-Euler 𝑙-function:𝑙^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(𝑠,𝑡,𝜒)=∑^𝐹−1_{𝑎1,…,𝑎ℎ=0(𝑎1+⋯+𝑎ℎ,𝑝)=1}𝜒(𝑎₁+⋯+𝑎_ℎ)𝐻^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(𝑠,𝑎₁,…,𝑎_ℎ,𝑝𝑡∣𝐹).(4.13)

Theorem 4.4. Let 𝜒 be a Dirichlet's character with an odd conductor 𝑑, and let 𝐹 be a positive multiple of 𝑝 and 𝑑. Then 𝑙^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑡,𝜒) is a 𝑝-adic analytic function on 𝐷 with𝑙^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑡,𝜒)=𝐸^(ℎ)_{𝑚,𝑞,𝜉,𝜒𝑚}𝐹(𝑝𝑡)−^𝑚[𝐹]^ℎ_−𝑞𝜒_𝑚(𝑝)_𝛽∈𝐼0𝜒_𝑚(𝛽)(−𝜉^𝑝𝑞^𝑝)^𝛽𝐸^(ℎ)_{𝑚,𝑞𝐹,𝜉𝐹}(𝛽+𝑡𝐹/𝑝),(4.14)where 𝑚∈ℤ₊ and𝐼₀1={𝑝𝑥|𝑥=𝑎₁+⋯+𝑎_ℎ≡0(mod𝑝)forsome𝑎₁,…,𝑎_ℎwith0≤𝑎₁,…,𝑎_ℎ≤𝐹−1},(4.15) and in ∑_𝛽∈𝐼0 one sums over 𝛽=(1/𝑝)𝑥 as many times as 𝑥 is expressed in the form 𝑥=𝑎₁+⋯+𝑎_ℎ by various 𝑎_𝑗's, and 𝜒_𝑚=𝜒𝜔^−𝑚 with 𝜔 the Teichmüller in the sense of the product of characters.Remark 4.5. Theorem 4.4 can be extended to obtain similar results for the multiple 𝑝-adic 𝐿-function in [16]. In the case ℎ=1, we note that_𝛽∈𝐼0=^{𝐹/𝑝−1}_𝑎=0.(4.16)Observe that if ℎ=1, thenlim_ℎ→1𝑙^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑡,𝜒)=𝐸_{𝑚,𝑞,𝜉,𝜒𝑚}𝐹(𝑝𝑡)−^𝑚[𝐹]_−𝑞𝜒_𝑚(𝑝)^{𝐹/𝑝−1}_𝑎=0𝜒_𝑚(𝑎)(−𝜉^𝑝𝑞^𝑝)^𝑎𝐸_{𝑚,𝑞𝐹,𝜉}^𝐹(𝑎+𝑡)𝐹/𝑝=𝐸_{𝑚,𝑞,𝜉,𝜒𝑚}(𝑝𝑡)−𝑝^𝑚𝜒_𝑚(𝑝)[𝐹/𝑝]_−𝑞𝑝[𝐹]_−𝑞(𝐹𝑝)^𝑚1[𝐹/𝑝]_−𝑞𝑝^{𝐹/𝑝−1}_𝑎=0𝜒_𝑚(𝑎)(−𝜉^𝑝𝑞^𝑝)^𝑎𝐸_{𝑚,(𝑞𝑝)}^𝐹/𝑝_,(𝜉^𝑝₎^𝐹/𝑝(𝑎+𝑡)(𝐹/𝑝whereweuseProposition2.3)=𝐸_{𝑚,𝑞,𝜉,𝜒𝑚}(𝑝𝑡)−𝑝^𝑚1[𝑝]_−𝑞𝜒_𝑚(𝑝)𝐸_{𝑚,𝑞𝑝,𝜉}^𝑝_,𝜒𝑚(𝑡).(4.17)This function interpolates the twisted generalized 𝑞-Euler polynomials at negative integers. For 𝑙⁽¹⁾_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑡,𝜒), the twisted 𝑝-adic 𝑞-Euler 𝑙-functions similar results were obtained (cf. see for detail [8, Theorem 9]). If 𝑞→1 in the above, thenlim_𝑞→1𝑙⁽¹⁾_{𝑝,𝑞,𝜉,𝐸}(𝑠,𝑡,𝜒)=𝑙_{𝑝,𝜉,𝐸}(𝑠,𝑡,𝜒)=2^∞_{𝑙=0(𝑙,𝑝)=1}𝜒(𝑙)(−1)^𝑙𝜉^𝑙(𝑡+𝑙)^𝑠,|𝑡|_𝑝≤1,(4.18)which is called the twisted 𝑝-adic 𝑙_𝐸-function of two variables.Proof. The formula for 𝑙^(ℎ)_{𝑞,𝜉,𝐸}(𝑠,𝑥,𝜒) is 𝑝-adic analytic function in 𝐷 by the Theorem 4.2. On the other hand, by substituting 𝑠=−𝑚,𝑚∈ℤ₊, into Definition 4.3, we have𝑙^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑡,𝜒)=^𝐹−1_{𝑎1,…,𝑎ℎ=0(𝑎1+⋯+𝑎ℎ,𝑝)=1}𝜒(𝑎₁+⋯+𝑎_ℎ)𝐻^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ=,𝑝𝑡∣𝐹)_{𝐹−1𝑎1,…,𝑎ℎ=0}𝜒(𝑎₁+⋯+𝑎_ℎ)𝐻^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ−,𝑝𝑡∣𝐹)_{𝐹−1𝑎1,…,𝑎ℎ=0(𝑎1+⋯+𝑎ℎ,𝑝)≠1}𝜒(𝑎₁+⋯+𝑎_ℎ)𝐻^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ,𝑝𝑡∣𝐹).(4.19)From Theorems 2.2, 3.1, and 4.2, we obtain^𝐹−1_{𝑎1,…,𝑎ℎ=0(𝑎1+⋯+𝑎ℎ,𝑝)≠1}𝜒(𝑎₁+⋯+𝑎_ℎ)𝐻^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑎₁,…,𝑎_ℎ=,𝑝𝑡∣𝐹)_{𝐹−1𝑎1,…,𝑎ℎ=0(𝑎1+⋯+𝑎ℎ,𝑝)≠1}𝜒_𝑚(𝑎₁+⋯+𝑎_ℎ)(−𝜉𝑞)^{𝑎1+⋯+𝑎ℎ}𝐹^𝑚[𝐹]^ℎ_−𝑞𝐸^(ℎ)_{𝑚,𝑞𝐹,𝜉𝐹}(𝑎₁+⋯+𝑎_ℎ+𝑝𝑡𝐹)=𝐹^𝑚[𝐹]^ℎ_−𝑞𝜒_𝑚(𝑝)_𝛽∈𝐼0𝜒_𝑚(𝛽)(−𝜉^𝑝𝑞^𝑝)^𝛽𝐸^(ℎ)_{𝑚,𝑞𝐹,𝜉𝐹}(𝛽+𝑡𝐹/𝑝).(4.20)Therefore,𝑙^(ℎ)_{𝑝,𝑞,𝜉,𝐸}(−𝑚,𝑡,𝜒)=𝐸^(ℎ)_{𝑚,𝑞,𝜉,𝜒𝑚}𝐹(𝑝𝑡)−^𝑚[𝐹]^ℎ_−𝑞𝜒_𝑚(𝑝)_𝛽∈𝐼0𝜒_𝑚(𝛽)(−𝜉^𝑝𝑞^𝑝)^𝛽𝐸^(ℎ)_{𝑚,𝑞𝐹,𝜉𝐹}(𝛽+𝑡𝐹/𝑝).(4.21)This completes the proof.

Acknowledgment

This work is supported by Kyungnam University Foundation Grant, 2008.