A note on Carlitz q-Bernoulli numbers and polynomials
© Kim and Kim; licensee Springer. 2012
Received: 21 December 2011
Accepted: 13 April 2012
Published: 13 April 2012
In this article, we first aim to give simple proofs of known formulae for the generalized Carlitz q-Bernoulli polynomials βm,χ(x, q) in the p-adic case by means of a method provided by Kim and then to derive a complex, analytic, two-variable q-L-function that is a q-analog of the two-variable L-function. Using this function, we calculate the values of two-variable q-L-functions at nonpositive integers and study their properties when q tends to 1.
Mathematics Subject Classification (2000): 11B68; 11S80.
Let p be a fixed prime. We denote by ℤ p , ℚ p , and ℂ p the ring of p-adic integers, the field of p-adic numbers, and the completion of the algebraic closure of ℚ p , respectively. Let v p be the normalized exponential valuation of ℂ p with . When one talks of a q-extension, q can be variously considered as an indeterminate, a complex number q ∈ ℂ, or a p-adic number q ∈ ℂ p . If q ∈ ℂ, one normally assumes |q| < 1. If q ∈ ℂ p , one normally assumes |1 - q| p < p-1/(p-1), so that q x = exp(x log p q) for |x| p ≤ 1.
Hence limq→1 [x] q = x for any x ∈ ℂ in the complex case and any x with |x| p ≤ 1 in the present p-adic case. This is the hallmark of a q-analog: The limit as q → 1 recovers the classical object.
In 1954, Carlitz  generalized a result of Frobenius  and showed many of the properties of the q-Bernoulli numbers β m (q). In 1964, Carlitz  extended the Bernoulli, Eulerian, and Euler numbers and corresponding polynomials as a formal Dirichlet series. In what follows, we shall call them the Carlitz q-Bernoulli numbers and polynomials.
Some properties of Carlitz q-Bernoulli numbers β m (q) were investigated by various authors. In , Koblitz constructed a q-analog of p-adic L-functions and suggested two questions. Question (1) was solved by Satoh . He constructed a complex analytic q-L-series that is a q-analog of Dirichlet L-function and interpolates Carlitz q-Bernoulli numbers, which is an answer to Koblitz's question. By using a q-analog of the p-adic Haar distribution (see (1.6) below), Kim  answered part of Koblitz's question (2) and constructed q-analogs of the p-adic log gamma functions Gp,q(x) on ℂ p \ ℤ p .
where q is a complex number with 0 < |q| < 1. He could not explicitly determine F q (t) in ℂ p , see [11, p.347].
where m ≥ 0 and q ∈ ℂ p with .
where q ∈ ℂ p with .
This article is organized as follows.
In Section 2, we consider the generalized Carlitz q-Bernoulli polynomials in the p-adic case by means of a method provided by Kim. We obtain the generating functions of the generalized Carlitz q-Bernoulli polynomials. We shall provide some basic formulas for the generalized Carlitz q-Bernoulli polynomials which will be used to prove the main results of this article.
In Section 3, we construct the complex, analytic, two-variable q-L-function that is a q- analog of the two-variable L-function. Using this function, we calculate the values of two-variable q-L-functions at nonpositive integers and study their properties when q tends to 1.
2. Generalized Carlitz q-Bernoulli polynomials in the p-adic (and complex) case
For any uniformly differentiable function f : ℤ p → ℂ p , the p-adic q-integral on ℤ p is defined to be the limit as N → ∞. The uniform differentiability guarantees the limit exists. Kim [12, 14–16] introduced this construction, denoted I q (f), where |1 - q| p < p-1/(p-1).
The construction of I q (f) makes sense for many q in ℂ p with the weaker condition |1 - q| p < 1. Indeed, when |1 - q| p < 1 the function q x is uniformly differentiable and the space of uniformly differentiable functions ℤ p → ℂ p is closed under multiplication, so we can make sense of its p-adic q-integral I q (f) for |1 - q| p < 1.
This completes the proof.
Remark 2.3. If χ = χ0, the trivial character and x = 0, then (2.1) reduces to (1.7) since d = 1. In particular, Kim  defined a class of p-adic interpolation functions G p,q (x) of the Carlitz q-Bernoulli numbers β m (q) and gave several interesting applications of these functions.
By Lemma 2.1, we can prove the following explicit formula of βm,χ(x, q) in ℂ p .
This completes the proof.
Remark 2.5. We note here that similar expressions to those of Proposition 2.4 with χ = χ0 are given by Kamano [, Proposition 2.6] and Kim [12, §2]. Also, Ryoo et al. [19, Theorem 4] gave the explicit formula of βm,χ(0, q) in ℂ for m ≥ 0.
Proof. If we write m = ad + k, where 0 ≤ k ≤ d - 1 and a = 0,1, 2,..., we have the desired result.
for each z ≠ 0 ∈ ℤ and p > 3, we find 0 < |q| < 1, 0 < |1 - q| p < 1.
Then, by (2.4), (2.5), (2.6), and (2.7), we have the following theorem.
Remark 2.8. If χ = χ 0, the trivial character, and x = 0, (2.8) reduces to (1.5).
3. q-analog of the two-variable L-function (in ℂ)
Hence we can define a q-analog of the L-function as follows:
In particular, the two-variable function L q (s, x, χ) is a generalization of the one-variable L q (s, χ) of Satoh , yielding the one-variable function when the second variable vanishes.
Proof. The proof is clear by Proposition 2.4, Theorem 2.7 and (3.1).
The formula of Proposition 3.2 is slight extension of the result in  and [11, Theorem 2].
where the Bk,χ(x) are the kth generalized Bernoulli polynomials.
Remark 3.5. The formula of Theorem 3.3 is slight extension of the result in [, Theorem 1].
This work was supported by the Kyungnam University Foundation Grant, 2012.
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