Open Access

A note on Carlitz q-Bernoulli numbers and polynomials

Advances in Difference Equations20122012:44

DOI: 10.1186/1687-1847-2012-44

Received: 21 December 2011

Accepted: 13 April 2012

Published: 13 April 2012

Abstract

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.

Keywords

Carlitz q-Bernoulli numbers Carlitz q-Bernoulli polynomials Dirichlet q-L- functions

1. Introduction

Let p be a fixed prime. We denote by p , p , and p the ring of p-adic integers, the field of p-adic numbers, and the completion of the algebraic closure of p , respectively. Let v p be the normalized exponential valuation of p with p p = p - v p ( p ) = p - 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_IEq1_HTML.gif. When one talks of a q-extension, q can be variously considered as an indeterminate, a complex number q , or a p-adic number q p . If q , one normally assumes |q| < 1. If q p , one normally assumes |1 - q| p < p-1/(p-1), so that q x = exp(x log p q) for |x| p ≤ 1.

Let d be a fixed positive integer. Let
X = X d = lim N ( / d p N ) , X 1 = p , X * = 0<a<dp ( a , p ) = 1 a + d p p , a + d p N p = { x X | x a ( mod d p N ) } , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ1_HTML.gif
(1.1)
where a lies in 0 ≤ a < dp N . We use the following notation:
[ x ] q = 1 - q x 1 - q . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ2_HTML.gif
(1.2)

Hence limq→1 [x] q = x for any x 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 [1] and, in 1941, Carlitz [2] discussed generalized Bernoulli and Euler numbers. Since that time, many authors have studied these and other related subjects (see, e.g., [36]). The final breakthrough came in the 1948 article by Carlitz [7]. He defined inductively new q-Bernoulli numbers β m = β m (q) by
β 0 ( q ) = 1 , q ( q β ( q ) + 1 ) m - β m ( q ) = 1 if m = 1 0 if m > 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ3_HTML.gif
(1.3)
with the usual convention of β i by β i . The q-Bernoulli polynomials are defined by
β m ( x , q ) = ( q x β ( q ) + [ x ] q ) m = i = 0 m m i β i ( q ) q i x [ x ] q m - i . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ4_HTML.gif
(1.4)

In 1954, Carlitz [8] generalized a result of Frobenius [3] and showed many of the properties of the q-Bernoulli numbers β m (q). In 1964, Carlitz [9] 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 [10], Koblitz constructed a q-analog of p-adic L-functions and suggested two questions. Question (1) was solved by Satoh [11]. 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 [12] answered part of Koblitz's question (2) and constructed q-analogs of the p-adic log gamma functions Gp,q(x) on p \ p .

In [11], Satoh constructed the generating function of the Carlitz q-Bernoulli numbers F q (t) in which is given by
F q ( t ) = m = 0 q m e [ m ] q t ( 1 - q - q m t ) = m = 0 β m ( q ) t m m ! , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ5_HTML.gif
(1.5)

where q is a complex number with 0 < |q| < 1. He could not explicitly determine F q (t) in p , see [11, p.347].

In [12], Kim defined the q-analog of the p-adic Haar distribution μHaar(a + p N p ) = 1/p N by
μ q ( a + p N p ) = q a [ p N ] q . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ6_HTML.gif
(1.6)
Using this distribution, he proved that the Carlitz q-Bernoulli numbers β m (q) can be represented as the p-adic q-integral on p by μ q , that is,
β m ( q ) = p [ a ] q m d μ q ( a ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ7_HTML.gif
(1.7)
and found the following explicit formula
β m ( q ) = 1 ( q - 1 ) m i = 0 m ( - 1 ) m - i m i i + 1 [ i + 1 ] q , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ8_HTML.gif
(1.8)

where m ≥ 0 and q p with 0 < 1 - q p < p - 1 p - 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_IEq2_HTML.gif.

Recently, Kim and Rim [13] constructed the generating function of the Carlitz q-Bernoulli numbers F q (t) in p :
F q ( t ) = e t 1 - q j = 0 j + 1 [ j + 1 ] q ( - 1 ) j 1 1 - q j t j j ! , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ9_HTML.gif
(1.9)

where q p with 0 < 1 - q p < p - 1 p - 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_IEq2_HTML.gif.

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 : p p , the p-adic q-integral on p is defined to be the limit 1 [ p N ] q a = 0 p N - 1 f ( a ) q a https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_IEq3_HTML.gif as N → ∞. The uniform differentiability guarantees the limit exists. Kim [12, 1416] 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 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 p 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 p with 0 < |1 - q| p < 1 and x p , we have
lim N 1 1 - q p N a = 0 p N - 1 q a x = x 1 - q x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equa_HTML.gif
Proof. We assume that q p satisfies the condition 0 < |1 - q| p < 1. Then it is known that
q x = m = 0 x m ( q - 1 ) m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equb_HTML.gif
for any x p (see [[17], Lemma 3.1 (iii)]). Therefore, we obtain
lim N 1 1 - q p N a = 0 p N - 1 q a x = 1 1 - q x lim N q p N x - 1 q p N - 1 = 1 1 - q x lim N m = 1 x m q p N - 1 m q p N - 1 = 1 1 - q x lim N m = 0 x m + 1 q p N - 1 m = x 1 - q x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equc_HTML.gif

This completes the proof.

Definition 2.2 ([12, §2, p. 323]). Let χ be a primitive Dirichlet character with conductor d and let x p . For q 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
β m , χ ( x , q ) = X χ ( a ) [ x + a ] q m d μ q ( a ) = lim N 1 [ d p N ] q a = 0 d p N - 1 χ ( a ) [ x + a ] q m q a . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ10_HTML.gif
(2.1)

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 [12] 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 p .

Proposition 2.4. For q p with 0 < |1 - q| p < 1 and an integer m ≥ 0, we have
β m , χ ( x , q ) = 1 ( 1 - q ) m k = 0 d - 1 χ ( k ) q k i = 0 m m i ( - 1 ) i q i ( x + k ) i + 1 [ d ( i + 1 ) ] q . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equd_HTML.gif
Proof. For m ≥ 0, (2.1) implies
β m , χ ( x , q ) = lim N 1 [ d ] q 1 [ p N ] q d k = 0 d - 1 a = 0 p N - 1 χ ( k + d a ) [ x + k + d a ] q m q k + d a = lim N 1 ( 1 - q ) m - 1 k = 0 d - 1 χ ( k ) q k 1 - q d p N q = 0 p N - 1 ( 1 - q x + k + d a ) m q d a = 1 ( 1 - q ) m - 1 k = 0 d - 1 χ ( k ) q k i = 0 m m i ( - 1 ) i q i ( x + k ) × lim N 1 1 - ( q d ) p N a = 0 p N - 1 q d a ( i + 1 ) = 1 ( 1 - q ) m - 1 k = 0 d - 1 χ ( k ) q k i = 0 m m i ( - 1 ) i q i ( x + k ) i + 1 1 - q d ( i + 1 ) (where we use Lemma 2 .1). https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Eque_HTML.gif

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 [[18], Proposition 2.6] and Kim [12, §2]. Also, Ryoo et al. [19, Theorem 4] gave the explicit formula of βm,χ(0, q) in for m ≥ 0.

Lemma 2.6. Let χ be a primitive Dirichlet character with conductor d . Then for q with |q| < 1,
m = 0 χ ( m ) q m x = 1 1 - q d x k = 0 d - 1 χ ( k ) q k x . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equf_HTML.gif

Proof. If we write m = ad + k, where 0 ≤ kd - 1 and a = 0,1, 2,..., we have the desired result.

We now consider the case:
q ¯ p , 0 < q < 1 , 0 < 1 - q p < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ11_HTML.gif
(2.2)
For instance, if we set
q = 1 1 - p z ¯ p https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equg_HTML.gif

for each z ≠ 0 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
F q , χ ( t , x ) = m = 0 β m , χ ( x , q ) t m m ! = m = 0 1 ( 1 - q ) m k = 0 d - 1 χ ( k ) q k i = 0 m m i ( - 1 ) i q i ( x + k ) i + 1 [ d ( i + 1 ) ] q t m m ! = P q , χ ( t , x ) + Q q , χ ( t , x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ12_HTML.gif
(2.3)
where
P q , χ ( t , x ) = m = 0 1 ( 1 - q ) m k = 0 d - 1 χ ( k ) q k i = 0 m m i ( - 1 ) i q i ( x + k ) i [ d ( i + 1 ) ] q t m m ! https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equh_HTML.gif
and
Q q , χ ( t , x ) = m = 0 1 ( 1 - q ) m k = 0 d - 1 χ ( k ) q k i = 0 m m i ( - 1 ) i q i ( x + k ) 1 [ d ( i + 1 ) ] q t m m ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equi_HTML.gif
Then, noting that
e t 1 - q = i = 0 ( - 1 ) i ( q - 1 ) - i t i i ! , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equj_HTML.gif
we see that
P q , χ ( t , x ) = m = 0 1 ( 1 - q ) m k = 0 d - 1 χ ( k ) q k i = 0 m m i ( - 1 ) i q i ( x + k ) i [ d ( i + 1 ) ] q t m m ! = n = 0 1 ( 1 - q ) n t n n ! j = 0 1 ( q - 1 ) j k = 0 d - 1 χ ( k ) q j ( x + k ) + k j [ d ( j + 1 ) ] q t j j ! = e t 1 - q j = 0 1 q - 1 j k = 0 d - 1 χ ( k ) q j ( x + k ) + k j [ d ( j + 1 ) ] q t j j ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ13_HTML.gif
(2.4)
Moreover, (2.4) now becomes
P q , χ ( t , x ) = e t 1 - q j = 1 1 q - 1 j k = 0 d - 1 χ ( k ) q j ( x + k ) + k 1 [ d ( j + 1 ) ] q t j ( j - 1 ) ! = e t 1 - q j = 0 1 q - 1 j q ( j + 1 ) x k = 0 d - 1 χ ( k ) q k ( j + 2 ) q d ( j + 2 ) - 1 t j + 1 j ! = - t e t 1 - q j = 0 1 q - 1 j q ( j + 1 ) x n = 0 χ ( n ) q n ( j + 2 ) t j j ! (where we use Lemma 2 .6) = - t e t 1 - q n = 0 χ ( n ) q x + 2 n j = 0 - q n + x 1 - q j t j j ! = - t e t 1 - q n = 0 χ ( x ) q x + 2 n e ( - q n + x ) t 1 - q = - t n = 0 χ ( x ) q x + 2 n e [ n + x ] q t https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ14_HTML.gif
(2.5)
(cf. [13, 16, 20]). Similar arguments apply to the case Qq,χ(t, x). We can rewrite
Q q , χ ( t , x ) = e t 1 - q j = 0 1 q - 1 j k = 0 d - 1 χ ( k ) q j ( x + k ) + k 1 [ d ( j + 1 ) ] q t j j ! https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ15_HTML.gif
(2.6)
and
Q q , χ ( t , x ) = ( 1 - q ) n = 0 χ ( n ) q n e [ n + x ] q t . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ16_HTML.gif
(2.7)

Then, by (2.4), (2.5), (2.6), and (2.7), we have the following theorem.

Theorem 2.7. Let q ¯ p , 0 < q < 1 , 0 < 1 - q p < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_IEq4_HTML.gif. 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:
F q , χ ( t , x ) = e t 1 - q j = 0 1 q - 1 j - 1 k = 0 d - 1 χ ( k ) q j ( x + k ) + k j + 1 q d ( j + 1 ) - 1 t j j ! = n = 0 χ ( n ) q n e [ n + x ] q t ( 1 - q - q n + x t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ17_HTML.gif
(2.8)

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 )

From Theorem 2.7, for k ≥ 0, we obtain the following
β k , χ ( x , q ) = d d t k F q , χ ( t , x ) t = 0 = ( 1 - q ) m = 0 χ ( m ) q m [ m + x ] q k - k m = 0 χ ( m ) q x + 2 m [ m + x ] q k - 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ18_HTML.gif
(3.1)

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 . 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) × defined by
L q ( s , x , χ ) = 1 - q s - 1 m = 0 χ ( m ) q m [ m + x ] q s - 1 + m = 0 χ ( m ) q m + 2 x [ m + x ] q s https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ19_HTML.gif
(3.2)

for 0 < x ≤ 1 (cf. [11, 13, 14, 2125]).

In particular, the two-variable function L q (s, x, χ) is a generalization of the one-variable L q (s, χ) of Satoh [11], yielding the one-variable function when the second variable vanishes.

Proposition 3.2. For k , k ≥ 1, the limiting value limskL q (1 - s, x, χ) = L q (1 - k, x, χ) exists and is given explicitly by
L q ( 1 - k , x , χ ) = - 1 k β k , χ ( x , q ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equk_HTML.gif

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 [19] and [11, Theorem 2].

Theorem 3.3. For any positive integer k, we have
lim q 1 β k , χ ( x , q ) = lim q 1 1 ( 1 - q ) m k = 0 d - 1 χ ( k ) q k i = 0 m m i ( - 1 ) i q i ( x + k ) i + 1 [ d ( i + 1 ) ] q = B k , χ ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equl_HTML.gif

where the Bk,χ(x) are the kth generalized Bernoulli polynomials.

Proof. We follow the proof in [[26], 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
i q i - 1 q i x = 1 log q i log q e i log q - 1 e x ( i log q ) = 1 log q k = 0 B k ( x ) i k ( log q ) k k ! , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ20_HTML.gif
(3.3)
where it is noted that in this instance, the notation B k (x) is used to replace B k (x) symbolically. For each m ≥ 1, let
( e t - 1 ) m = k = 0 d k ( m ) t k k ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ21_HTML.gif
(3.4)
Note that
( e t - 1 ) m = i = 0 m ( - 1 ) m - i m i e i t = k = 0 i = 0 m m i ( - 1 ) m - i i k t k k ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ22_HTML.gif
(3.5)
From (3.4) and (3.5), we obtain
d k ( m ) = i = 0 m ( - 1 ) m - i m i i k , m k 0 , 0 k < m . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equ23_HTML.gif
(3.6)
It is also clear from the definition that d 0 ( 0 ) = 1 , d k ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_IEq5_HTML.gif and d k ( k ) = k ! https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_IEq6_HTML.gif for k . From (2.3), (3.3), and (3.6), we obtain
β m , χ ( x , q ) = q - x ( q - 1 ) m k = 0 d - 1 χ ( k ) q k + x i = 0 m ( - 1 ) m - i m i q i ( k + x ) i + 1 d ( i + 1 ) q = q - x ( q - 1 ) m - 1 k = 0 d - 1 χ ( k ) i = 0 m ( - 1 ) m - i m i × e d ( i + 1 ) log q ( k + x ) d d ( i + 1 ) log q e d ( i + 1 ) log q - 1 1 d log q = q - x ( q - 1 ) m - 1 n = 0 i = 0 m ( - 1 ) m - i m i ( i + 1 ) n × d n - 1 k = 0 d - 1 χ ( k ) B n k + x d ( log q ) n - 1 n ! = q - x ( log q ) m - 1 ( q - 1 ) m - 1 d m - 1 k = 0 d - 1 χ ( k ) B m k + x d + q - x σ = 1 i = 0 σ m + σ i d m + σ - i ( m ) 1 ( m + σ ) ! ( log q ) m + σ - 1 ( q - 1 ) m - 1 × d m + σ - 1 k = 0 d - 1 χ ( k ) B m + σ k + x d . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equm_HTML.gif
Then, because
log q = log ( 1 + ( q - 1 ) ) = ( q - 1 ) - ( q - 1 ) 2 2 + = ( q - 1 ) + O ( ( q - 1 ) 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equn_HTML.gif
as q → 1, we find
lim q 1 ( log q ) m + σ - 1 ( q - 1 ) m - 1 = 1 , σ = 0 0 , σ 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equo_HTML.gif
so
lim q 1 β m , χ ( x , q ) = d m - 1 k = 0 d - 1 χ ( k ) B m k + x d = B m , χ ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equp_HTML.gif

where the Bm,χ(x) are the m th generalized Bernoulli polynomials (e.g., [14, 19]). This completes the proof.

Corollary 3.4. For any positive integer k, we have
lim q 1 L q ( 1 - k , x , χ ) = - 1 k B k , x ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equq_HTML.gif

Remark 3.5. The formula of Theorem 3.3 is slight extension of the result in [[26], 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 [[27], p. 8]
F χ ( t , x ) : = lim q 1 F q , χ ( t , x ) = - t a = 1 d l = 0 χ ( a + d l ) e ( a + d l ) t e x t = a = 1 d χ ( a ) t e ( a + x ) t e d t - 1 = m = 0 B m , χ ( x ) t m m ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equr_HTML.gif
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
ζ ( s , x ) = m = 0 1 ( m + x ) s https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equs_HTML.gif
by
ζ q ( s , x ) = L q ( s , x , χ 0 ) = 1 - q s - 1 m = 0 q m + x [ m + x ] q s - 1 + m = 0 q 2 ( m + x ) [ m + x ] q s https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equt_HTML.gif
and obtain the identity
lim q 1 ζ q ( s , x ) = ζ ( s , x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equu_HTML.gif
for all s ≠ 1, as well as the formula
lim q 1 ζ q ( 1 - k , x ) = - 1 k B k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2012-44/MediaObjects/13662_2011_Article_175_Equv_HTML.gif

for integers k ≥ 1 (cf. [11, 13, 19, 22, 24, 25]).

Declarations

Acknowledgements

This work was supported by the Kyungnam University Foundation Grant, 2012.

Authors’ Affiliations

(1)
National Institute for Mathematical Sciences, Doryong-dong
(2)
Division of Cultural Education, Kyungnam University

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