On A two-variable p-adic l_q function

We prove that a two-variable p-adic l_q-function has the series p-adic expansion which interpolates a linear combinations of terms of the generalized q-Euler polynomials at non positive integers. The proof of this original construction is due to Kubota and Leopoldt in 1964, although the method given this note is due to Washington

which interpolates a linear combinations of terms of the generalized q-Euler polynomials at non positive integers. The proof of this original construction is due to Kubota and Leopoldt in 1964, although the method given this note is due to Washington.

Introduction
The ordinary Euler polynomials E n (t) are defined by the equation x n n! .
Setting t = 1/2 and normalizing by 2 n gives the ordinary Euler numbers E n = 2 n E n 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 [9], Kim et al. recently defined the natural q-extension of ordinary Euler numbers and polynomials by padic 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 that for n = 0, 1, . . . , where E * n,χω −n ,q is a generalized q-Euler numbers attached to the Dirichlet characters χω −n (see Section 2 for definitions).
In the present paper, we shall construct a specific two-variable p-adic l q -function l p,q (s, t, χ) by means of a method provided in [13,3,8]. 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 a linear combinations of terms of the generalized q-Euler polynomials at non positive integers. 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. [3,5,6,7,8,9,11,12,13]).
Thought 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 completion of the algebraic closure of Q p , respectively. We will use Z + for the set of non positive 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: [5,6].
Let d be a fixed integer, and let where a ∈ Z lies in 0 ≤ a < dp N . Let U D(Z p ) be the space of uniformly differentiable function on Z p . For f ∈ U D(Z p ), the p-adic q-integral was defined by In [6], the bosonic integral was considered from a more physical point of view to the bosonic limit q → 1 as follows: Furthermore, we can consider the fermionic integral in contrast to the conventional "bosonic." That is, I −1 (f ) = Zp f (a)dµ −1 (a) (see [7]). From this, we derive where f n (a) = f (a+ n) and n ∈ Z + (see [7]). 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:

q-Euler numbers and polynomials
In this section, we review some notations and facts in [9]. From (1.4), we can derive the following formula: Hence we obtain We now set E * n,q is called q-Euler numbers. By (2.2) and (2.3), we see that Zp a n dµ −q (a) = E * n,q . From (2.2), we also note that In view of (2.3) and (2.4), we can consider q-Euler polynomials associated to t as follows: x n n! and where E n and E n (t) are the ordinary Euler numbers and polynomials. By (2.3) and (2.5), we easily see that E * n,q (t) = n m=0 , see [9].
If d is odd positive integer, we have Let χ be a Dirichlet character with conductor In view of (2.8), we also consider the generalized q-Euler polynomials attached to χ as follows: x n n! .
We write n = a + dl, where n = 1, 2, . . . , and obtain Note that l q (s, t, χ) is an analytic function in the whole complex s-plane. By using a geometric series in (2.9), we obtain x n n! .
We also note that By Definition 2.1 and (2.16), we obtain the following theorem.
Proposition 2.3. For n ∈ Z + , we have l q (−n, t, χ) = E * n,χ,q (t). These values of l q (s, t, χ) at netative 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.

A two-variable p-adic l q -function
We shall consider the p-adic analogue 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 exists 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 (mod pZ 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 (mod pZ 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 as If t ∈ C p such that |t| p ≤ 1, then for any a ∈ Z, a + pt = a (mod pR) 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 to the character χ n : (Z/lcm(d, p)Z) × → C × defined by χ n (a) = χ(a)ω −n (a). 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, we define the two-variable p-adic l q -functions as follows: 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 [3,13] can be the functions ∞ m=0 s m (F/(a + pt)) m E * m,q F and a + pt s = ∞ m=0 s m ( a + pt − 1) m is analytic for s ∈ D. According to this method, we see that the function ∞ m=0 s m (F/(a + pt)) m E * m,q F 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 −1 pt) m is analytic for t ∈ C p , |t| p ≤ 1, when s ∈ D. Therefore, l q (s, t, χ) is analytic for t ∈ C p , |t| p ≤ 1, provided s ∈ D (see [3]).
We set Thus,we note that We also consider the two-variable p-adic l q -functions which interpolate the generalized q-Euler polynomials at negative integers as follows: χ(a)h p,q (s, t, a|F ).
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 d = d χ . Then for each n ∈ Z, n ≥ 0, We can derive by a manipulation of an appropriate generating functions. Set χ n = χω −n . From (3.2) and (3.3), we obtain for n ∈ Z + . From Lemma 3.2, we see that admits an analytic function for t ∈ C p with |t| p ≤ 1 and s ∈ D, and satisfies the relation l p,q (−n, t, χ) = E * n,χn,q (pt) − p n χ n (p) [2] q [2] q p E * n,χn,q p (t) for n ∈ Z + and t ∈ C p with |t| p ≤ 1 From (3.3) 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.

Then
(1) l p (s, t, χ) is analytic for t ∈ C p with |t| p ≤ 1 and s ∈ D.