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A note on Carlitz q-Bernoulli numbers and polynomials
Advances in Difference Equations volume 2012, Article number: 44 (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 qx= exp(x log p q) for |x| p ≤ 1.
Let d be a fixed positive integer. Let
where a ∈ ℤ lies in 0 ≤ a < dpN. We use the following notation:
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 1937, Vandiver  and, in 1941, Carlitz  discussed generalized Bernoulli and Euler numbers. Since that time, many authors have studied these and other related subjects (see, e.g., [3–6]). The final breakthrough came in the 1948 article by Carlitz . He defined inductively new q-Bernoulli numbers β m = β m (q) by
with the usual convention of βiby β i . The q-Bernoulli polynomials are defined by
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 .
In , Satoh constructed the generating function of the Carlitz q-Bernoulli numbers F q (t) in ℂ which is given by
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].
In , Kim defined the q-analog of the p-adic Haar distribution μHaar(a + pNℤ p ) = 1/pNby
Using this distribution, he proved that the Carlitz q-Bernoulli numbers β m (q) can be represented as the p-adic q-integral on ℤ p by μ q , that is,
and found the following explicit formula
where m ≥ 0 and q ∈ ℂ p with .
Recently, Kim and Rim  constructed the generating function of the Carlitz q-Bernoulli numbers F q (t) in ℂ p :
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 qxis 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.
Lemma 2.1. For q ∈ ℂ p with 0 < |1 - q| p < 1 and x ∈ ℤ p , we have
Proof. We assume that q ∈ ℂ p satisfies the condition 0 < |1 - q| p < 1. Then it is known that
for any x ∈ ℤ p (see [, Lemma 3.1 (iii)]). Therefore, we obtain
This completes the proof.
Definition 2.2 ([12, §2, p. 323]). Let χ be a primitive Dirichlet character with conductor d ∈ ℕ and let x ∈ ℤ p . For q ∈ ℂ p with 0 < |1 - q| p < 1 and an integer m ≥ 0, the generalized Carlitz q-Bernoulli polynomials βm,χ(x, q) are defined by
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 .
Proposition 2.4. For q ∈ ℂ p with 0 < |1 - q| p < 1 and an integer m ≥ 0, we have
Proof. For m ≥ 0, (2.1) implies
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.
Lemma 2.6. Let χ be a primitive Dirichlet character with conductor d ∈ ℕ. Then for q ∈ ℂ with |q| < 1,
Proof. If we write m = ad + k, where 0 ≤ k ≤ d - 1 and a = 0,1, 2,..., we have the desired result.
We now consider the case:
For instance, if we set
for each z ≠ 0 ∈ ℤ and p > 3, we find 0 < |q| < 1, 0 < |1 - q| p < 1.
Let Fq,χ(t, x) be the generating function of βm,χ(x, q) defined in Definition 2.2. From Proposition 2.4, we have
Then, noting that
we see that
Moreover, (2.4) now becomes
Then, by (2.4), (2.5), (2.6), and (2.7), we have the following theorem.
Theorem 2.7. Let. Then the generalized Carlitz q-Bernoulli polynomials βm,χ(x, q) for m ≤ 0 is given by equating the coefficients of powers of t in the following generating function:
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)
From Theorem 2.7, for k ≥ 0, we obtain the following
Hence we can define a q-analog of the L-function as follows:
Definition 3.1. Suppose that χ is a primitive Dirichlet character with conductor d ∈ ℕ. Let q be a complex number with 0 < |q| < 1, and let L q (s, x, χ) be a function of two-variable (s, x) ∈ ℂ × ℝ defined by
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.
Proposition 3.2. For k ∈ ℤ, k ≥ 1, the limiting value lims → kL q (1 - s, x, χ) = L q (1 - k, x, χ) exists and is given explicitly by
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].
Theorem 3.3. For any positive integer k, we have
where the Bk,χ(x) are the kth generalized Bernoulli polynomials.
Proof. We follow the proof in [, Theorem 1] motivated by the study of a simple q-analog of the Riemann zeta function. Recall that the ordinary Bernoulli polynomials B k (x) are defined by
where it is noted that in this instance, the notation B k (x) is used to replace Bk(x) symbolically. For each m ≥ 1, let
From (3.4) and (3.5), we obtain
It is also clear from the definition that and for k ∈ ℕ. From (2.3), (3.3), and (3.6), we obtain
as q → 1, we find
Corollary 3.4. For any positive integer k, we have
Remark 3.5. The formula of Theorem 3.3 is slight extension of the result in [, Theorem 1].
Remark 3.6. From Theorem 2.7, the generalized Bernoulli polynomials Bm,χ(x) are defined by means of the following generating function [, p. 8]
Remark 3.7. If we substitute χ = χ0, the trivial character, in Definition 3.1 and Corollary 3.4, we can also define a q-analog of the Hurwitz zeta function
and obtain the identity
for all s ≠ 1, as well as the formula
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This work was supported by the Kyungnam University Foundation Grant, 2012.
The authors declare that they have no competing interests.
The authors have equal contributions to each part of this paper. All authors read and approved the final manuscript.
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Kim, D., Kim, MS. A note on Carlitz q-Bernoulli numbers and polynomials. Adv Differ Equ 2012, 44 (2012). https://doi.org/10.1186/1687-1847-2012-44
- Carlitz q-Bernoulli numbers
- Carlitz q-Bernoulli polynomials
- Dirichlet q-L- functions