Skip to main content

Theory and Modern Applications

A note on Carlitz q-Bernoulli numbers and polynomials

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.

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 . 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 qx= 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 ) } ,
(1.1)

where a lies in 0 ≤ a < dpN. We use the following notation:

[ x ] q = 1 - q x 1 - q .
(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 ,
(1.3)

with the usual convention of βiby 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 .
(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 ! ,
(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 + pN p ) = 1/pNby

μ q ( a + p N p ) = q a [ p N ] q .
(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 ) ,
(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 ,
(1.8)

where m ≥ 0 and q p with 0< 1 - q p < p - 1 p - 1 .

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 ! ,
(1.9)

where q p with 0< 1 - q p < p - 1 p - 1 .

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 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 qxis 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 .

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

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 .

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 .
(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 .

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).

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 .

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 .
(2.2)

For instance, if we set

q = 1 1 - p z ¯ p

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 ) ,
(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 !

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 ! .

Then, noting that

e t 1 - q = i = 0 ( - 1 ) i ( q - 1 ) - i t i i ! ,

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 ! .
(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
(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 !
(2.6)

and

Q q , χ ( t , x ) = ( 1 - q ) n = 0 χ ( n ) q n e [ n + x ] q t .
(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 . 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 ) .
(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 .
(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
(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 ) .

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 ) ,

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 ! ,
(3.3)

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

( e t - 1 ) m = k = 0 d k ( m ) t k k ! .
(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 ! .
(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 .
(3.6)

It is also clear from the definition that d 0 ( 0 ) =1, d k ( 0 ) =0 and d k ( k ) =k! 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 .

Then, because

log q = log ( 1 + ( q - 1 ) ) = ( q - 1 ) - ( q - 1 ) 2 2 + = ( q - 1 ) + O ( ( q - 1 ) 2 )

as q → 1, we find

lim q 1 ( log q ) m + σ - 1 ( q - 1 ) m - 1 = 1 , σ = 0 0 , σ 1 ,

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 ) ,

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 ) .

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 ! .

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

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

and obtain the identity

lim q 1 ζ q ( s , x ) = ζ ( s , x )

for all s ≠ 1, as well as the formula

lim q 1 ζ q ( 1 - k , x ) = - 1 k B k ( x )

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

References

  1. Vandiver HS: On generalizations of the numbers of Bernoulli and Euler. Proc Natl Acad Sci USA 1937, 23(10):555–559.

    Article  Google Scholar 

  2. Carlitz L: Generalized Bernoulli and Euler numbers. Duke Math J 1941, 8: 585–589.

    Article  MathSciNet  Google Scholar 

  3. Frobenius G: Uber die Bernoullischen Zahlen und die Eulerschen Polynome. Preuss Akad Wiss Sitzungsber 1910, 809–847.

    Google Scholar 

  4. Nielsen N: Traité Élémentaire des Nombres de Bernoulli. Gauthier-Villars, Paris; 1923.

    Google Scholar 

  5. Nörlund NE: Vorlesungen über Differentzenrechnung. Springer-Verlag, Berlin (1924) (Reprinted by Chelsea Publishing Company, Bronx, New York; 1954.

    Google Scholar 

  6. Vandiver HS: On general methods for obtaining congruences involving Bernoulli numbers. Bull Am Math Soc 1940, 46: 121–123.

    Article  MathSciNet  Google Scholar 

  7. Carlitz L: q -Bernoulli numbers and polynomials. Duke Math J 1948, 15: 987–1000.

    Article  MathSciNet  Google Scholar 

  8. Carlitz L: q -Bernoulli and Eulerian numbers. Trans Am Soc 1954, 76: 332–350.

    Google Scholar 

  9. Carlitz L: Extended Bernoulli and Eulerian numbers. Duke Math J 1964, 31: 667–689.

    Article  MathSciNet  Google Scholar 

  10. Koblitz N: On Carlitz's q -Bernoulli numbres. J Number Theory 1982, 14: 332–339.

    Article  MathSciNet  Google Scholar 

  11. Satoh J: q -Analogue of Riemann's ζ -function and q -Euler numbers. J Number Theory 1989, 31: 346–362.

    Article  MathSciNet  Google Scholar 

  12. Kim T: On a q -analogue of the p- adic log gamma functions and related integrals. J Number Theory 1999, 76: 320–329.

    Article  MathSciNet  Google Scholar 

  13. Kim T, Rim SH: A note on p- adic Carlitz q -Bernoulli numbers. Bull Austral Math 2000, 62: 227–234.

    Article  MathSciNet  Google Scholar 

  14. Kim T: On explicit formulas of p -adic q - L -functions. Kyushu J Math 1994, 48: 73–86.

    Article  MathSciNet  Google Scholar 

  15. Kim T: A note on some formulae for the q-Euler numbers and polynomials. Proc Jangjeon Math Soc 2006, 9: 227–232.

    MathSciNet  Google Scholar 

  16. Kim T: q -Bernoulli Numbers Associated with q -Stirling Numbers. Adv Diff Equ 2008, 2008: 10. Article ID 743295

    Article  Google Scholar 

  17. Conrad K: A q -analogue of Mahler expansions. I Adv Math 2000, 153: 185–230.

    Article  MathSciNet  Google Scholar 

  18. Kamano K: p -adic q -Bernoulli numbers and their denominators. Int J Number Theory 2008, 4: 911–925.

    Article  MathSciNet  Google Scholar 

  19. Ryoo CS, Kim T, Lee B: q -Bernoulli numbers and q -Bernoulli polynomials revisited. Adv Diff Equ 2011, 2011: 33.

    Article  Google Scholar 

  20. Rim SH, Bayad A, Moon EJ, Jin JH, Lee SJ: A new construction on the q-Bernoulli polynomials. Adv Diff Equ 2011, 2011: 34.

    Article  MathSciNet  Google Scholar 

  21. Cenkci M, Simsek Y, Kurt V: Further remarks on multiple p -adic q - L -function of two variables. Adv Stud Contemp Math 2007, 14: 49–68.

    MathSciNet  Google Scholar 

  22. Kim T: A note on the q-multiple zeta function. Adv Stud Contemp Math 2004, 8: 111–113.

    Google Scholar 

  23. Simsek Y: Twisted ( h, q )-Bernoulli numbers and polynomials related to twisted ( h, q )-zeta function and L-function. J Math Anal Appl 2006, 324: 790–804.

    Article  MathSciNet  Google Scholar 

  24. Simsek Y: On twisted q -Hurwitz zeta function and q -two-variable L -function. Appl Math Comput 2007, 187: 466–473.

    Article  MathSciNet  Google Scholar 

  25. Simsek Y: On p -adic twisted q - L -functions related to generalized twisted Bernoulli numbers. Russian J Math Phys 2006, 13: 340–348.

    Article  MathSciNet  Google Scholar 

  26. Kaneko M, Kurokawa N, Wakayama M: A variation of Euler's approach to values of the Riemann zeta function. Kyushu Math J 2003, 57: 175–192.

    Article  MathSciNet  Google Scholar 

  27. Iwasawa K: Lectures on p -adic L -functions. In Ann Math Studies. Volume 74. Princeton, New Jersey; 1972.

    Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Min-Soo Kim.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The authors have equal contributions to each part of this paper. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1847-2012-44

Keywords