On a Two-Variable 𝑝-Adic 𝑙_𝑞-Function

Abstract

We prove that a two-variable 𝑝-adic 𝑙_𝑞-function has the series expansion 𝑙_𝑝,𝑞(𝑠,𝑡,𝜒)=([2]_𝑞/[2]_𝐹)∑^𝐹_{𝑎=1,(𝑝,𝑎)=1}(−1)^𝑎(𝜒(𝑎)𝑞^𝑎/⟨𝑎+𝑝𝑡⟩^𝑠)∑^∞_𝑚=0(_𝑚^−𝑠)(𝐹/⟨𝑎+𝑝𝑡⟩)^𝑚𝐸^∗_{𝑚,𝑞𝐹} which interpolates the values 𝑙_𝑝,𝑞(−𝑛,𝑡,𝜒)=𝐸^∗_{𝑛,𝜒𝑛,𝑞}(𝑝𝑡)−𝑝^𝑛𝜒_𝑛(𝑝)([2]_𝑞/[2]_𝑞𝑝)𝐸^∗_{𝑛,𝜒𝑛,𝑞𝑝}(𝑡), whenever 𝑛 is a nonpositive integer. The proof of this original construction is due to Kubota and Leopoldt in 1964, although the method given in this note is due to Washington.

1. Introduction

The ordinary Euler polynomials 𝐸_𝑛(𝑡) are defined by the equation2𝑒^𝑡𝑥𝑒^𝑥=+1^∞_𝑛=0𝐸_𝑛𝑥(𝑡)^𝑛.𝑛!(1.1)Setting 𝑡=1/2 and normalizing by 2^𝑛 gives the ordinary Euler numbers𝐸_𝑛=2^𝑛𝐸_𝑛(12).(1.2)The ordinary Euler polynomials appear in many classical results (see [1]). In [2], the values of these polynomials at rational arguments were expressed in term of the Hurwitz zeta function. Congruences for Euler numbers have also received much attention from the point of view of 𝑝-adic interpolation. In [3], Kim et al. recently defined the natural 𝑞-extension of ordinary Euler numbers and polynomials by 𝑝-adic integral representation and proved properties generalizing those satisfied by 𝐸_𝑛 and 𝐸_𝑛(𝑡). They also constructed the one-variable 𝑝-adic 𝑞-𝑙-function 𝑙_𝑝,𝑞(𝑠,𝜒) for Dirichlet characters 𝜒 and 𝑠∈ℂ_𝑝 with |𝑠|_𝑝<𝑝^{1−(1/(𝑝−1))}, with the property that𝑙_𝑝,𝑞(−𝑛,𝜒)=𝐸^∗_{𝑛,𝜒𝜔−𝑛,𝑞}−[2]_𝑞[2]_𝑞⁻¹_𝑝𝑝^𝑛𝜒𝜔^−𝑛(𝑝)𝐸^∗_{𝑛,𝜒𝜔−𝑛,𝑞𝑝}(1.3)for 𝑛=0,1,…, where 𝐸^∗_{𝑛,𝜒𝜔−𝑛,𝑞} is a generalized 𝑞-Euler number associated with the Dirichlet characters 𝜒𝜔^−𝑛 (see Section 2 for definitions).

In the present paper, we will construct a specific two-variable 𝑝-adic 𝑙_𝑞-function 𝑙_𝑝,𝑞(𝑠,𝑡,𝜒) by means of a method provided in [4–6]. We also prove that 𝑙_𝑝,𝑞(𝑠,𝑡,𝜒) is analytic in 𝑠 and 𝑡 for 𝑠∈ℂ_𝑝 with |𝑠|_𝑝<𝑝^{1−(1/(𝑝−1))} and 𝑡∈ℂ_𝑝 with |𝑡|_𝑝≤1, which interpolates the values𝑙_𝑝,𝑞(−𝑛,𝑡,𝜒)=𝐸^∗_{𝑛,𝜒𝑛,𝑞}(𝑝𝑡)−𝑝^𝑛𝜒_𝑛(𝑝)[2]_𝑞[2]_𝑞𝑝𝐸^∗_{𝑛,𝜒𝑛,𝑞𝑝}(𝑡),(1.4)whenever 𝑛 is a nonpositive integer. This two-variable function is a generalization of the one-variable 𝑝-adic 𝑞-𝑙-function, which is the function obtained by putting 𝑡=0 in 𝑙_𝑝,𝑞(𝑠,𝑡,𝜒) (cf. [3–11]).

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. We will use ℤ⁺ for the set of nonpositive integers. 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. 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)^𝑥1+𝑞,(cf.[7,8]).(1.5) Let 𝑑 be a fixed integer, and let𝑋=𝑋_𝑑=lim_←𝑁(ℤ/𝑑𝑝^𝑁ℤ),𝑋^∗=⋃_{0<𝑎<𝑑𝑝(𝑎,𝑝)=1}𝑎+𝑑𝑝ℤ_𝑝,𝑎+𝑑𝑝^𝑁ℤ_𝑝={𝑥∈𝑋∣𝑥≡𝑎(mod𝑑𝑝^𝑁)},(1.6)where 𝑎∈ℤ lies in 0≤𝑎<𝑑𝑝^𝑁. Let 𝑈𝐷(ℤ_𝑝) be the space of uniformly differentiable function on ℤ_𝑝. For 𝑓∈𝑈𝐷(ℤ_𝑝), the 𝑝-adic 𝑞-integral is defined by𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝑎)𝑑𝜇_𝑞(𝑎)=_𝑋𝑓(𝑎)𝑑𝜇_𝑞(𝑎)=lim_𝑁→∞1[𝑑𝑝^𝑁]_𝑞^{𝑑𝑝𝑁−1}_𝑎=0𝑓(𝑎)𝑞^𝑎for|1−𝑞|_𝑝<1.(1.7)In [8], the bosonic integral was considered from a more physical point of view to the limit 𝑞→1 as follows:𝐼₁(𝑓)=lim_𝑞→1𝐼_𝑞(𝑓)=_ℤ𝑝𝑓(𝑎)𝑑𝜇₁(𝑎)=lim_𝑁→∞1𝑝^{𝑁𝑝𝑁−1}_𝑎=0𝑓(𝑎).(1.8)Furthermore, we can consider the fermionic integral in contrast to the conventional “bosonic integral.” That is, 𝐼₋₁(𝑓)=∫_ℤ𝑝𝑓(𝑎)𝑑𝜇₋₁(𝑎) (see [9]). From this, we derive 𝐼₋₁(𝑓₁)+𝐼₋₁(𝑓)=2𝑓(0), where 𝑓₁(𝑎)=𝑓(𝑎+1). Also, we have𝐼₋₁(𝑓_𝑛)+(−1)^𝑛−1𝐼₋₁(𝑓)=2^𝑛−1_𝑎=0(−1)^{𝑛−1−𝑎}𝑓(𝑎),(1.9)where 𝑓_𝑛(𝑎)=𝑓(𝑎+𝑛) and 𝑛∈ℤ⁺ (see [9]). 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(cf.[3]).(1.10)

2. 𝑞-Euler Numbers and Polynomials

In this section, we review some notations and facts in [3].

From (1.10), we can derive the following formula:𝑞𝐼_−𝑞(𝑓₁)+𝐼_−𝑞(𝑓)=[2]_𝑞𝑓(0),(2.1)where 𝑓₁(𝑎) is translation with 𝑓₁(𝑎)=𝑓(𝑎+1). If we take 𝑓(𝑎)=𝑒^𝑎𝑥, then we have 𝑓₁(𝑎)=𝑒^(𝑎+1)𝑥=𝑒^𝑎𝑥𝑒^𝑥. From (2.1), we derive (𝑞𝑒^𝑥+1)𝐼_−𝑞(𝑒^𝑎𝑥)=[2]_𝑞. Hence, we obtain𝐼_−𝑞(𝑒^𝑎𝑥)=_ℤ𝑝𝑒^𝑎𝑥𝑑𝜇_−𝑞(𝑎)=[2]_𝑞𝑞𝑒^𝑥.+1(2.2)We now set[2]_𝑞𝑞𝑒^𝑥=+1^∞_𝑛=0𝐸^∗_𝑛,𝑞𝑥^𝑛.𝑛!(2.3)𝐸^∗_𝑛,𝑞 is called the 𝑛th 𝑞-Euler number. By (2.2) and (2.3), we see that_ℤ𝑝𝑎^𝑛𝑑𝜇_−𝑞(𝑎)=𝐸^∗_𝑛,𝑞.(2.4)From (2.2), we also note that_ℤ𝑝𝑒^{(𝑡+𝑎)𝑥}𝑑𝜇_−𝑞(𝑎)=[2]_𝑞𝑞𝑒^𝑥𝑒+1^𝑡𝑥.(2.5)In view of (2.3) and (2.5), we can consider 𝑞-Euler polynomials associated with 𝑡 as follows:[2]_𝑞𝑞𝑒^𝑥𝑒+1^𝑡𝑥=^∞_𝑛=0𝐸^∗_𝑛,𝑞𝑥(𝑡)^𝑛,𝑛!_ℤ𝑝(𝑡+𝑎)^𝑛𝑑𝜇_−𝑞(𝑎)=𝐸^∗_𝑛,𝑞(𝑡).(2.6)Put lim_𝑞→1𝐸^∗_𝑛,𝑞=𝐸^∗_𝑛 and lim_𝑞→1𝐸^∗_𝑛,𝑞(𝑡)=𝐸^∗_𝑛(𝑡). Then, we have 𝐸_𝑛(𝑡)=𝐸^∗_𝑛(𝑡) and𝐸_𝑛=^𝑛_𝑚=02^𝑚𝑛𝑚𝐸^∗_𝑚,(2.7)where 𝐸_𝑛 and 𝐸_𝑛(𝑡) are the ordinary Euler numbers and polynomials. By (2.3) and (2.6), we easily see that 𝐸^∗_𝑛,𝑞(𝑡)=∑^𝑛_𝑚=0(^𝑛_𝑚)𝑡^𝑛−𝑚𝐸^∗_𝑚,𝑞. For 𝑑∈ℤ⁺, let 𝑓_𝑑(𝑎)=𝑓(𝑎+𝑑). Then, we have𝑞^𝑑𝐼_−𝑞(𝑓_𝑑)+(−1)^𝑑−1𝐼_−𝑞(𝑓)=[2]_𝑞^𝑑−1_𝑎=0(−1)^{𝑑−𝑎−1}𝑞^𝑎𝑓(𝑎),see[3].(2.8)If 𝑑 is an odd positive integer, we have𝑞^𝑑𝐼_−𝑞(𝑓_𝑑)+𝐼_−𝑞(𝑓)=[2]_𝑞^𝑑−1_𝑎=0(−1)^𝑎𝑞^𝑎𝑓(𝑎).(2.9)Let 𝜒 be a Dirichlet character with conductor 𝑑=𝑑_𝜒 (=odd) ∈ℤ⁺. If we take 𝑓(𝑎)=𝜒(𝑎)𝑒^{(𝑡+𝑎)𝑥}, then we have 𝑓_𝑑(𝑎)=𝑓(𝑎+𝑑)=𝜒(𝑎)𝑒^𝑑𝑥𝑒^{(𝑡+𝑎)𝑥}. From (1.7) and (2.9), we derive_𝑋𝜒(𝑎)𝑒^{(𝑡+𝑎)𝑥}𝑑𝜇_−𝑞(𝑎)=[2]_𝑞∑^𝑑_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)𝑒^{(𝑡+𝑎)𝑥}𝑞^𝑑𝑒^𝑑𝑥.+1(2.10)In view of (2.10), we also consider the generalized 𝑞-Euler polynomials associated with 𝜒 as follows:𝐹_𝜒,𝑞(𝑥,𝑡)=[2]_𝑞∑^𝑑_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)𝑒^{(𝑡+𝑎)𝑥}𝑞^𝑑𝑒^𝑑𝑥=+1^∞_𝑛=0𝐸^∗_{𝑛,𝜒,𝑞}𝑥(𝑡)^𝑛.𝑛!(2.11)From (2.10) and (2.11), we derive the following equation:_𝑋𝜒(𝑎)(𝑡+𝑎)^𝑛𝑑𝜇_−𝑞(𝑎)=𝐸^∗_{𝑛,𝜒,𝑞}(𝑡)(2.12)for 𝑛≥0. Put lim_𝑞→1𝐸^∗_{𝑛,𝜒,𝑞}(𝑡)=𝐸^∗_𝑛,𝜒(𝑡). On the other hand, the generalized 𝑞-Euler polynomials associated with 𝜒 are easily expressed as the 𝑞-Euler polynomials:𝐸^∗_{𝑛,𝜒,𝑞}(𝑡)=𝑑^𝑛[2]_𝑞[2]_𝑞𝑑^𝑑_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)𝐸^∗_{𝑛,𝑞𝑑}(𝑎+𝑡𝑑),𝑛≥0.(2.13) Let 𝜒 be a Dirichlet character with conductor 𝑑=𝑑_𝜒∈ℤ⁺. It is well known (see [11, 12]) that, for positive integers 𝑚 and 𝑛,^𝑑𝑛_𝑎=1𝜒(𝑎)𝑎^𝑚=1𝑚+1(𝐵_𝑚+1,𝜒(𝑑𝑛)−𝐵_𝑚+1,𝜒(0)),(2.14)where 𝐵_𝑚+1,𝜒(𝑡) are the generalized Bernoulli polynomials. When 𝑑=𝑑_𝜒 (=odd) ∈ℤ⁺, note that[2]_𝑞∑^𝑑_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)𝑒^{(𝑡+𝑎)𝑥}(1−(−𝑞^𝑑𝑒^𝑑𝑥)^𝑛)1−(−𝑞^𝑑𝑒^𝑑𝑥)=[2]_𝑞^𝑑_𝑎=1^𝑛−1_𝑙=0(−1)^{𝑎+𝑑𝑙}𝑞^{𝑎+𝑑𝑙}𝜒(𝑎+𝑑𝑙)𝑒^{𝑥(𝑡+𝑎+𝑑𝑙)}=[2]_𝑞^𝑑𝑛_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)𝑒^{𝑥(𝑡+𝑎)}=^∞_𝑚=0([2]_𝑞^𝑑𝑛_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)(𝑡+𝑎)^𝑚)𝑥^𝑚.𝑚!(2.15)By (2.11), the relation (2.15) can be rewritten as[2]_𝑞∑^𝑑_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)𝑒^{(𝑡+𝑎)𝑥}(1−(−𝑞^𝑑𝑒^𝑑𝑥)^𝑛)1−(−𝑞^𝑑𝑒^𝑑𝑥)=^∞_𝑚=0(𝐸^∗_{𝑚,𝜒,𝑞}(𝑡)+(−1)^𝑛+1𝑞^𝑑𝑛𝐸^∗_{𝑚,𝜒,𝑞}𝑥(𝑡+𝑑𝑛))^𝑚.𝑚!(2.16)Now, we give the 𝑞-analog of (2.14) for the generalized Euler polynomials. From (2.15) and (2.16), it is easy to see that^𝑑𝑛_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)(𝑡+𝑎)^𝑚=1[2]_𝑞(𝐸^∗_{𝑚,𝜒,𝑞}(𝑡)+(−1)^𝑛+1𝑞^𝑑𝑛𝐸^∗_{𝑚,𝜒,𝑞}(𝑡+𝑑𝑛))(2.17)for positive integers 𝑚 and 𝑛. In particular, replacing 𝑞 by 1 in (2.17), if 𝜒=𝜒⁰, the principal character (𝑑_𝜒=1), and 𝑡=0, then^𝑛−1_𝑎=1(−1)^𝑎𝑎^𝑚=12(𝐸_𝑚(0)+(−1)^𝑛+1𝐸_𝑚(𝑛)).(2.18)Definition 2.1. Let 𝑠∈ℂ with Re(𝑠)>1. Let 𝜒 be a primitive Dirichlet character with conductor 𝑑=𝑑_𝜒 (=odd) ∈ℤ⁺. One sets𝑙_𝑞(𝑠,𝑡,𝜒)=[2]_𝑞^∞_𝑛=0(−1)^𝑛𝑞^𝑛𝜒(𝑛)(𝑡+𝑛)^𝑠,0<𝑡≤1.(2.19)Remark 2.2. We assume that 𝑞∈ℂ with |𝑞|<1. Let 𝜒 be a primitive Dirichlet character with conductor 𝑑=𝑑_𝜒 (=odd) ∈ℤ⁺. From (2.11), we consider the below integral which is known as the Mellin transformation of 𝐹_𝜒,𝑞(𝑥,𝑡) (cf. [13])1Γ(𝑠)^∞₀𝑥^𝑠−1𝐹_𝜒,𝑞(−𝑥,𝑡)𝑑𝑥=[2]_𝑞^𝑑_𝑎=1(−1)^𝑎𝑞^𝑎1𝜒(𝑎)Γ(𝑠)^∞₀𝑥^𝑠−1𝑒^{−(𝑡+𝑎)𝑥}1−(−𝑞^𝑑𝑒^−𝑑𝑥)𝑑𝑥=[2]_𝑞^𝑑_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎+𝑑𝑙)^∞_𝑙=0(−1)^𝑑𝑙𝑞^𝑑𝑙(𝑎+𝑑𝑙+𝑡)^𝑠.(2.20)We write 𝑛=𝑎+𝑑𝑙, where 𝑛=1,2,…, and obtain1Γ(𝑠)^∞₀𝑥^𝑠−1𝐹_𝜒,𝑞(−𝑥,𝑡)𝑑𝑥=[2]_𝑞^∞_𝑛=0(−1)^𝑛𝑞^𝑛𝜒(𝑛)(𝑡+𝑛)^𝑠=𝑙_𝑞(𝑠,𝑡,𝜒).(2.21)

Note that 𝑙_𝑞(𝑠,𝑡,𝜒) is an analytic function in the whole complex 𝑠-plane. By using a geometric series in (2.11), we obtain[2]_𝑞𝑒^∞𝑡𝑥_𝑛=0(−1)^𝑛𝑞^𝑛𝜒(𝑛)𝑒^𝑛𝑥=^∞_𝑛=0𝐸^∗_{𝑛,𝜒,𝑞}𝑥(𝑡)^𝑛.𝑛!(2.22)We also note that𝐸^∗_{𝑛,𝜒,𝑞}(𝑡)=(dd𝑥)^𝑘[2]_𝑞𝑒^∞𝑡𝑥_𝑛=0(−1)^𝑛𝑞^𝑛𝜒(𝑛)𝑒^𝑛𝑥|_𝑥=0.(2.23)By Definition 2.1 and (2.23), we obtain the following proposition.Proposition 2.3. For 𝑛∈ℤ⁺, one has 𝑙_𝑞(−𝑛,𝑡,𝜒)=𝐸^∗_{𝑛,𝜒,𝑞}(𝑡).

The values of 𝑙_𝑞(𝑠,𝑡,𝜒) at negative integers are algebraic, hence may be regarded as being in an extension of ℚ_𝑝. We therefore look for a 𝑝-adic function which agrees with 𝑙_𝑞(𝑠,𝑡,𝜒) at the negative integers in Section 3.

3. A Two-Variable 𝑝-Adic 𝑙_𝑞-Function

We will consider the 𝑝-adic analog of the 𝑙_𝑞-functions which are introduced in the previous section (see Definition 2.1). Throughout this section we assume that 𝑝 is an odd prime. Note that there exist 𝜑(𝑝) distinct solutions, modulo 𝑝, to the equation 𝑥^𝜑(𝑝)−1=0, and each solution must be congruent to one of the values 𝑎∈ℤ, where 1≤𝑎<𝑝,(𝑎,𝑝)=1. Thus, given 𝑎∈ℤ with (𝑎,𝑝)=1, there exists a unique 𝜔(𝑎)∈ℤ_𝑝, where 𝜔(𝑎)^𝜑(𝑝)=1, such that 𝜔(𝑎)≡𝑎(mod𝑝ℤ_𝑝). Letting 𝜔(𝑎)=0 for 𝑎∈ℤ, such that (𝑎,𝑝)≠1, it can be seen that 𝜔 is actually a Dirichlet character having conductor 𝑑_𝜔=𝑝, called the Teichmüller character. Let ⟨𝑎⟩=𝜔⁻¹(𝑎)𝑎. Then, ⟨𝑎⟩≡1(mod𝑝ℤ_𝑝). For the context in the sequel, an extension of the definition of the Teichmüller character is needed. We denote a particular subring of ℂ_𝑝 as𝑅={𝑎∈ℂ_𝑝∣|𝑎|_𝑝≤1}.(3.1)If 𝑡∈ℂ_𝑝, such that |𝑡|_𝑝≤1, then for any 𝑎∈ℤ,𝑎+𝑝𝑡≡𝑎(mod𝑝𝑅). Thus, for 𝑡∈ℂ_𝑝,|𝑡|_𝑝≤1,𝜔(𝑎+𝑝𝑡)=𝜔(𝑎). Also, for these values of 𝑡, let ⟨𝑎+𝑝𝑡⟩=𝜔⁻¹(𝑎)(𝑎+𝑝𝑡). Let 𝜒 be the Dirichlet character of conductor 𝑑=𝑑_𝜒. For 𝑛≥1, we define 𝜒_𝑛 to be the primitive character associated with the character 𝜒_𝑛∶(ℤ/lcm(𝑑,𝑝)ℤ)^×→ℂ^× defined by 𝜒_𝑛(𝑎)=𝜒(𝑎)𝜔^−𝑛(𝑎).

We define an interpolation function for generalized 𝑞-Euler polynomials.Definition 3.1. Let 𝜒 be the Dirichlet character with conductor 𝑑=𝑑_𝜒 (=odd) and let 𝐹 be a positive integral multiple of 𝑝 and 𝑑. Now, one defines the two-variable 𝑝-adic 𝑙_𝑞-function as follows:𝑙_𝑝,𝑞(𝑠,𝑡,𝜒)=[2]_𝑞[2]_𝑞𝐹^𝐹_{𝑎=1(𝑝,𝑎)=1}(−1)^𝑎𝜒(𝑎)𝑞^𝑎⟨𝑎+𝑝𝑡⟩^∞−𝑠_𝑚=0𝑚(−𝑠𝐹)⟨𝑎+𝑝𝑡⟩^𝑚𝐸^∗_{𝑚,𝑞𝐹}.(3.2)Let 𝐷={𝑠∈ℂ_𝑝∣|𝑠|_𝑝<𝑝^{1−(1/(𝑝−1))}} and let 𝑎∈ℤ,(𝑎,𝑝)=1. For 𝑡∈ℂ_𝑝,|𝑡|_𝑝≤1, the same argument as that given in the proof of the main theorem of [4, 5] can be used to show that the functions ∑^∞_𝑚=0(^𝑠_𝑚)(𝐹/(𝑎+𝑝𝑡))^𝑚𝐸^∗_{𝑚,𝑞𝐹} and ⟨𝑎+𝑝𝑡⟩^𝑠=∑^∞_𝑚=0(^𝑠_𝑚)(⟨𝑎+𝑝𝑡⟩−1)^𝑚 are analytic for 𝑠∈𝐷. According to this method, we see that the function ∑^∞_𝑚=0(^𝑠_𝑚)(𝐹/(𝑎+𝑝𝑡))^𝑚𝐸^∗_{𝑚,𝑞𝐹} is analytic for 𝑡∈ℂ_𝑝,|𝑡|_𝑝≤1, whenever 𝑠∈𝐷. It readily follows that ⟨𝑎+𝑝𝑡⟩^𝑠=⟨𝑎⟩^𝑠∑^∞_𝑚=0(^𝑠_𝑚)(𝑎⁻¹𝑝𝑡)^𝑚 is analytic for 𝑡∈ℂ_𝑝,|𝑡|_𝑝≤1, when 𝑠∈𝐷. Therefore,𝑙_𝑝,𝑞(𝑠,𝑡,𝜒)isanalyticfor𝑡∈ℂ_𝑝,|𝑡|_𝑝≤1,(3.3) provided 𝑠∈𝐷 (see [5]).

We setℎ_𝑝,𝑞(𝑠,𝑡,𝑎∣𝐹)=(−1)^𝑎𝑞^𝑎⟨𝑎+𝑝𝑡⟩^−𝑠[2]_𝑞[2]_𝑞𝐹^∞_𝑚=0𝑚(−𝑠𝐹)𝑎+𝑝𝑡^𝑚𝐸^∗_{𝑚,𝑞𝐹}.(3.4)Thus, we note thatℎ_𝑝,𝑞(−𝑛,𝑡,𝑎∣𝐹)=𝜔^−𝑛(𝑎)(−1)^𝑎𝑞^𝑎𝐹^𝑛[2]_𝑞[2]_𝑞𝐹𝐸^∗_{𝑛,𝑞𝐹}(𝑎+𝑝𝑡𝐹)(3.5)for 𝑛∈ℤ⁺. We also consider the two-variable 𝑝-adic 𝑙_𝑞-functions which interpolate the generalized 𝑞-Euler polynomials at negative integers as follows:𝑙_𝑝,𝑞(𝑠,𝑡,𝜒)=^𝐹_{𝑎=1(𝑝,𝑎)=1}𝜒(𝑎)ℎ_𝑝,𝑞(𝑠,𝑡,𝑎∣𝐹).(3.6)We will in the process derive an explicit formula for this function. Before we begin this derivation, we need the following result concerning generalized 𝑞-Euler polynomials.Lemma 3.2. Let 𝐹 be a positive integral multiple of 𝑝 and 𝑑=𝑑_𝜒. Then, for each 𝑛∈ℤ,𝑛≥0,𝐸^∗_{𝑛,𝜒,𝑞}(𝑡)=𝐹^𝑛[2]_𝑞[2]_𝑞𝐹^𝐹_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)𝐸^∗_{𝑛,𝑞𝐹}(𝑎+𝑡𝐹).(3.7)Proof. By (2.6) and (2.11), we note that𝐹_𝜒,𝑞(𝑥,𝑡)=[2]_𝑞[2]_𝑞𝑑^𝑑_𝑎∑₌₁(−1)^𝑎𝑞^𝑎𝜒(𝑎)∑^∞_𝑛=0𝐸^∗_{𝑛,𝑞𝑑}(𝑎+𝑡𝑑)(𝑑𝑥)^𝑛.𝑛!(3.8)Then, we have𝐸^∗_{𝑛,𝜒,𝑞}(𝑡)=𝑑^𝑛[2]_𝑞[2]_𝑞𝑑^𝑑_𝑎∑₌₁(−1)^𝑎𝑞^𝑎𝜒(𝑎)𝐸^∗_{𝑛,𝑞𝑑}(𝑎+𝑡𝑑).(3.9)On the other hand, if 𝐹=𝑑𝑝, then we get[2]_𝑞[2]_𝑞𝐹^𝐹_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)^∞_𝑛=0𝐸^∗_{𝑛,𝑞𝐹}(𝑎+𝑡𝐹)(𝐹𝑥)^𝑛𝑛!=[2]_𝑞^𝑑_𝑎=1^𝑝−1_𝑏=0(−1)^{𝑎+𝑏𝑑}𝑞^{𝑎+𝑏𝑑}𝑒𝜒(𝑎+𝑏𝑑)^{(𝑎+𝑏𝑑)𝑥}𝑒^𝑡𝑥𝑞^𝐹𝑒^𝐹𝑥+1=[2]_𝑞^𝑑_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)𝑒^{(𝑎+𝑡)𝑥𝑝−1}_𝑏=0(−1)^𝑏𝑑𝑞^𝑏𝑑𝑒^𝑏𝑑𝑥1𝑞^𝐹𝑒^𝐹𝑥=+1[2]_𝑞[2]_𝑞𝑑^𝑑_𝑎=1(−1)^𝑎𝑞^𝑎𝜒(𝑎)^∞_𝑛=0𝐸^∗_{𝑛,𝑞𝑑}(𝑎+𝑡𝑑)(𝑑𝑥)^𝑛.𝑛!(3.10)This completes the proof.

Set 𝜒_𝑛=𝜒𝜔^−𝑛. From (3.5) and (3.6), we obtain𝑙_𝑝,𝑞(−𝑛,𝑡,𝜒)=𝐹^𝑛[2]_𝑞[2]_𝑞𝐹^𝐹_{𝑎=1(𝑝,𝑎)=1}𝜒_𝑛(𝑎)(−1)^𝑎𝑞^𝑎𝐸^∗_{𝑛,𝑞𝐹}(𝑎+𝑝𝑡𝐹)=𝐹^𝑛[2]_𝑞[2]_𝑞𝐹^𝐹_𝑎=1𝜒_𝑛(𝑎)(−1)^𝑎𝑞^𝑎𝐸^∗_{𝑛,𝑞𝐹}(𝑎+𝑝𝑡𝐹)−𝐹^𝑛[2]_𝑞[2]_𝑞𝐹^𝐹/𝑝_𝑎=1𝜒_𝑛(𝑝𝑎)(−1)^𝑝𝑎𝑞^𝑝𝑎𝐸^∗_{𝑛,𝑞𝐹}(𝑝𝑎+𝑝𝑡𝐹).(3.11)for 𝑛∈ℤ⁺. From Lemma 3.2, we see that𝐸^∗_{𝑛,𝜒𝑛,𝑞}(𝑝𝑡)=𝐹^𝑛[2]_𝑞[2]_𝑞𝐹^𝐹_𝑎=1(−1)^𝑎𝑞^𝑎𝜒_𝑛(𝑎)𝐸^∗_{𝑛,𝑞𝐹}(𝑎+𝑝𝑡𝐹𝐸),_{∗𝑛,𝜒𝑛,𝑞𝑝}𝐹(𝑡)=(𝑝)^𝑛[2]_𝑞𝑝[2]_(𝑞𝑝)^{𝐹/𝑝𝐹/𝑝}_𝑎=1(−1)^𝑎(𝑞^𝑝)^𝑎𝜒_𝑛(𝑎)𝐸^∗_{𝑛,(𝑞𝑝)𝐹/𝑝}(𝑎+𝑡𝐹/𝑝).(3.12) From (3.3), (3.11), and (3.12), we obtain the following theorem.

Theorem 3.3. Let 𝐹 (=odd) be a positive integral multiple of 𝑝 and 𝑑_𝜒. Then, the two-variable 𝑝-adic 𝑙_𝑞-function𝑙_𝑝,𝑞(𝑠,𝑡,𝜒)=[2]_𝑞[2]_𝑞𝐹^𝐹_{𝑎=1(𝑝,𝑎)=1}(−1)^𝑎𝜒(𝑎)𝑞^𝑎⟨𝑎+𝑝𝑡⟩^∞−𝑠_𝑚=0𝑚(−𝑠𝐹)⟨𝑎+𝑝𝑡⟩^𝑚𝐸^∗_{𝑚,𝑞𝐹}(3.13)admits an analytic function for 𝑡∈ℂ_𝑝 with |𝑡|_𝑝≤1 and 𝑠∈𝐷 and satisfies the relation𝑙_𝑝,𝑞(−𝑛,𝑡,𝜒)=𝐸^∗_{𝑛,𝜒𝑛,𝑞}(𝑝𝑡)−𝑝^𝑛𝜒_𝑛(𝑝)[2]_𝑞[2]_𝑞𝑝𝐸^∗_{𝑛,𝜒𝑛,𝑞𝑝}(𝑡)(3.14)for 𝑛∈ℤ⁺ and 𝑡∈ℂ_𝑝 with |𝑡|_𝑝≤1.

From (3.5) and Theorem 3.3, it follows that ℎ_𝑝,𝑞(𝑠,𝑡,𝑎∣𝐹) is analytic for 𝑡∈ℂ_𝑝 with |𝑡|_𝑝≤1 and 𝑠∈𝐷.Remark 3.4. Let ⟨𝑎+𝑝𝑡⟩=𝜔⁻¹(𝑎)(𝑎+𝑝𝑡), and let 𝑡∈ℂ_𝑝 with |𝑡|_𝑝≤1 and 𝑠∈𝐷. Then the two-variable 𝑝-adic 𝑙_𝑞-function defined above is redefined by𝑙_𝑝,𝑞(𝑠,𝑡,𝜒)=_𝑋∗𝜒(𝑎)⟨𝑎+𝑝𝑡⟩^−𝑠𝑑𝜇_−𝑞(𝑎),comparedto[3,10].(3.15)Then, we have𝑙_𝑝,𝑞(−𝑛,𝑡,𝜒)=_𝑋𝜒_𝑛(𝑎)(𝑎+𝑝𝑡)^𝑛𝑑𝜇_−𝑞(𝑎)−_𝑋𝜒_𝑛(𝑝𝑎)(𝑝𝑎+𝑝𝑡)^𝑛𝑑𝜇_−𝑞(𝑝𝑎)_(2.10)=𝐸^∗_{𝑛,𝜒𝑛,𝑞}(𝑝𝑡)−𝑝^𝑛𝜒_𝑛(𝑝)[2]_𝑞[2]_𝑞𝑝𝐸^∗_{𝑛,𝜒𝑛,𝑞𝑝}(𝑡),(3.16)since 𝑋^∗=𝑋−𝑝𝑋 and [2]_𝑞𝑝𝑑𝜇_−𝑞(𝑝𝑎)=[2]_𝑞𝑑𝜇_−𝑞𝑝(𝑎).

If 𝑞→1 in Theorem 3.3 and Remark 3.4, we obtain the following corollary.Corollary 3.5. Let 𝐹 (=odd) be a positive integral multiple of 𝑝 and 𝑑_𝜒, and let the two-variable 𝑝-adic 𝑙-function𝑙_𝑝(𝑠,𝑡,𝜒)=^𝐹_{𝑎=1(𝑝,𝑎)=1}(−1)^𝑎𝜒(𝑎)⟨𝑎+𝑝𝑡⟩^∞−𝑠_𝑚=0𝑚(−𝑠𝐹)⟨𝑎+𝑝𝑡⟩^𝑚𝐸^∗_𝑚.(3.17)Then, (1)𝑙_𝑝(𝑠,𝑡,𝜒) is analytic for 𝑡∈ℂ_𝑝 with |𝑡|_𝑝≤1 and 𝑠∈𝐷.(2)𝑙_𝑝(−𝑛,𝑡,𝜒)=𝐸^∗_{𝑛,𝜒𝑛}(𝑝𝑡)−𝑝^𝑛𝜒_𝑛(𝑝)𝐸^∗_{𝑛,𝜒𝑛}(𝑡) for 𝑛∈ℤ⁺.(3)𝑙_𝑝(𝑠,𝑡,𝜒)=∫_𝑋∗𝜒(𝑎)⟨𝑎+𝑝𝑡⟩^−𝑠𝑑𝜇₋₁(𝑎) for 𝑡∈ℂ_𝑝 with |𝑡|_𝑝≤1 and 𝑠∈𝐷.

Acknowledgment

This work is supported by Kyungnam University Foundation grant, 2007.