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We give some interesting equation of p-adic q-integrals onZp.. From those p-adic q-integrals, we present a systemic study of some families of extended Carlitz type q-Bernoulli numbers an

Volume 2010, Article ID 575240, 17 pages

doi:10.1155/2010/575240

Research Article

Numbers and Polynomials Associated with

q-Volkenborn Integrals

T Kim,1 J Choi,1 B Lee,2 and C S Ryoo3

Seoul 139-701, Republic of Korea

Correspondence should be addressed to T Kim,tkkim@kw.ac.kr

Received 23 August 2010; Accepted 30 September 2010

Academic Editor: Alberto Cabada

Copyrightq 2010 T Kim et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

We give some interesting equation of p-adic q-integrals onZp From those p-adic q-integrals,

we present a systemic study of some families of extended Carlitz type q-Bernoulli numbers and polynomials in p-adic number field.

1 Introduction

Let p be a fixed prime number Throughout this paper, Z p,Qp,C, and Cpwill, respectively,

denote the ring of p-adic rational integer, the field of p-adic rational numbers, the complex

number field, and the completion of algebraic closure of Qp Let N be the set of natural numbers andZ  {0} ∪ N

Let ν pbe the normalized exponential valuation ofCpwith|p| p  p −ν p p  p−1 When

one talks of q-extension, q is considered as an indeterminate, a complex number q ∈ C, or

assume that|1 − q| p < 1 We use the notation

x q 1− q x

The q-factorial is defined as

n q! n q n − 1 q· · · 2q1q , 1.2

and the Gaussian q-binomial coefficient is defined by



n k



q

 n q!

n − k q!kq!  n q n − 1 q · · · n − k  1 q

see 1 Note that

lim

q → 1



n k



q





n k



 n n − 1 · · · n − k  1

From1.3, we easily see that



n  1 k



q





n

k − 1



q

 q k



n k



q

 q n1−k



n

k − 1



q





n k



q

see 2,3 For a fixed positive integer f, f, p  1, let

X  X f  lim←−

N

 Z

fp NZ



, X1 Zp ,

0<a<fp

a,p1





, a  fp NZpx ∈ X | x ≡ a

modfp N , 1.6

where a ∈ Z and 0 ≤ a < fp Nsee 1 14

We say that f is a uniformly differential function at a point a ∈ Z p and denote this

property by f ∈ UDZ p if the difference quotients

F f



x, y

 f x − f



y

have a limit l  f a as x, y → a, a For f ∈ UDZ p, let us begin with the expression

1

p N

q

p N−1

x0

f xq x

0≤x<pN

f xμ q



x  p NZp

representing a q-analogue of the Riemann sums for f, see 1 3,11–18 The integral of f

onZp is defined as the limitN → ∞ of the sums if exists The p-adic integral 

q-Volkenborn integral of f ∈ UDZp is defined by

I q



f





X

f xdμ q x 



Zp

f xdμ q x  lim

N−→∞

1

p N

q ≤x<p N

f xq x , 1.9

see 12 Carlitz’s q-Bernoull numbers β k,q can be defined recursively by β 0,q 1 and by the rule that

q

qβ∗ 1k − β∗

k,q

⎧

⎨

⎩

1, if k  1,

0, if k > 1, 1.10

with the usual convention of replacingβ∗i by β∗i,q,see 1 13

It is well known that

β∗n,q



Zp

x n

q dμ q x 



X

x n

q dμ q x, n ∈ Z,

β∗n,q x 



Zp

y  x n

q dμ q



y





X

y  x n

q dμ q



y

1.11

see 1, where β∗

n,q x are called the nth Carlitz’s q-Bernoulli polynomials see 1,12,13

Let χ be the Dirichlet’s character with conductor f ∈ N, then the generalized Carlitz’s

q-Bernoulli numbers attached to χ are defined as follows:

β∗n,χ,q



X

χ xx n

see 13 Recently, many authors have studied in the different several areas related to

q-theorysee 1 13 In this paper, we present a systemic study of some families of multiple

Carlitz’s type q-Bernoulli numbers and polynomials by using the integral equations of p-adic

From these equations, we give some interesting formulae for the higher-order Carlitz’s type

q-Bernoulli numbers and polynomials in the p-adic number field.

and Polynomials

In this section, we assume that q ∈ C pwith|1 − q| p < 1 We first consider the q-extension of

Bernoulli polynomials as follows:

∞

n0

β n,q x t n



Z q −y e xy qt dμ q



y

 −t∞

m0

e xm qt q xm 2.1

From2.1, we note that

β n,q x   1

1− qn

n

l0



n l



−q xl l

l q

  1

1− qn−1

n

l0



n l



−q xl



l

1− q l



  n

1− qn−1

n−1

l0



n − 1 l



q l1x

 1

1− q l1



−1l1

  −n

1− qn−1

∞

m0

q mx

n−1

l0



n − 1 l



q lxm

 −n ∞

m0

q mx x  m n−1

q

2.2

Note that

lim

q → 1 β n,q x  −n∞

m0

x  m n−1  B n x, 2.3

where B n x are called the n th ordinary Bernoulli polynomials In the special case, x  0,

β n,q 0  β n,q are called the n th q-Bernoulli numbers.

By2.2, we have the following lemma

Lemma 2.1 For n ≥ 0, one has

β n,q x 



Zp

q −y

x  y n

q dμ q



y

 −n ∞

m0

q mx x  m n−1

q

  1

1− qn

n

l0



n l



−q xl l

l q .

2.4

Now, one considers the q-Bernoulli polynomials of order r ∈ N as follows:

∞

n0

β n,q r x t n



Zp

· · ·



Zp

  

r times

q −x1···x re xx1···x rqt dμ q x1 · · · dμ q x r .

2.5

By2.5, one sees that

β n,q r x 



Zp

· · ·



Zp

  

r times

q −x1···x rx  x1 · · ·  x rn

q dμ q x1 · · · dμ q x r

  1

1− qn

n

l0



n l



−1l q xl



l

l q

r

.

2.6

In the special case, x  0, the sequence β n,q r 0  β r n,q is refereed to as the q-extension of Bernoulli numbers of order r For f ∈ N, one has

β r n,q x 



X

· · ·



X

  

r times

q −x1···x rx  x1 · · ·  x rn

q dμ q x1 · · · dμ q x r

  1

1− qn

n

l0



n l



−1l f−1

a1, ,a r0

q lxa1···a r l r

lf r q

f n−r q f−1

a1, ,a r0

β r n,q f



a1 · · ·  a r  x

f



.

2.7

By2.5 and 2.7, one obtains the following theorem

Theorem 2.2 For r ∈ Z, f ∈ N, one has

β r n,q x   1

1− qn

n

l0

f−1

a1, ,a r0



n l



−1l q la1···a r x l r

lf r

q

f n−r

q f−1

a1, ,a r0

β r n,q f



a1 · · ·  a r  x

f



.

2.8

Let χ be the primitive Dirichlet’s character with conductor f ∈ N, then the generalized q-Bernoulli polynomials attached to χ are defined by

∞

n0

β n,χ,q x t n



X

χ

y

q −y e xy qt dμ q



y

From2.9, one derives

β n,χ,q x 



X

χ

y

q −y

x  y n

q dμ q



y



f−1

a0

χ a lim

N−→∞

1

fp N

q

fp N−1

y0

a  x  fy n

q

  1

1− qn f−1

a0

χ a n

l0



n l



−1l q lxa l

lf

q



f−1

a0

m0



−nx  a  mf n−1

q

 −n∞

m0

χ mx  m n−1

q

2.10

By2.9 and 2.10, one can give the generating function for the generalized q-Bernoulli polynomials

attached to χ as follows:

F χ,q x, t  −t ∞

m0

χ me xm qt  ∞

n0

β n,χ,q x t n

From1.3, 2.10, and 2.11, one notes that

β n,χ,q x  1

f

q

f−1

a0

χ a



Zp

q −fy

a  x  fy n

q dμ q f

y

f n−1

q f−1

a0

χ aβ n,q f



a  x f



.

2.12

In the special case, x  0, the sequence β n,χ,q 0  β n,χ,q are called the n th generalized q-Bernoulli numbers attached to χ.

Let one consider the higher-order q-Bernoulli polynomials attached to χ as follows:



X

· · ·



X

  

r times

 r



i1

χ x i



e xx1···x rqt q −x1···x rdμ q x1 · · · dμ q x r  ∞

n0

β r n,χ,q x t n

n! , 2.13

where β r n,χ,q x are called the n th generalized q-Bernoulli polynomials of order r attaches to χ.

By2.13, one sees that

β r n,χ,q x   1

1− qn

n

l0



n l



−q xl f−1

a1, ,a r0

 r



i1

χ a i



q lr i1 a i l r

lf r

q

f n−r

q f−1

a1, ,a r0

r

i1

χ a i



β r n,q f



x  a1 · · ·  a r

f



.

2.14

In the special case, x  0, the sequence β r n,χ,q 0  β r n,χ,q are called the n th generalized q-Bernoulli numbers of order r attaches to χ.

By2.13 and 2.14, one obtains the following theorem

Theorem 2.3 Let χ be the primitive Dirichlet’s character with conductor f ∈ N For n ∈ Z, r ∈ N, one has

β r n,χ,q x   1

1− qn

n

l0



n l



−q xl f−1

a1, ,a r0

r

i1

χ a i



q lr i1 a i l r

lf r

q

f n−r

q f−1

a1, ,a r0

 r



i1

χ a i



β r n,q f



x  a1 · · ·  a r

f



.

2.15

For h ∈ Z, and r ∈ N, one introduces the extended higher-order q-Bernoulli polynomials as follows:

β n,q h,r x 



Zp

· · ·



Zp

  

r times

qr j1 h−j−1x j x  x1 · · ·  x rn

q dμ q x1 · · · dμ q x r .

2.16

From2.16, one notes that

β n,q h,r x   1

1− qn

n

l0



n l



−1l q lx

lh−1

r



lh−1

r



q

r!

r q!, 2.17

and

β h,r n,q x f n−r

q f−1

a1, ,a r0

qr j1 h−ja j β h,r n,q f



x  a1 · · ·  a r

f



. 2.18

In the special case, x  0, β n,q h,r 0  β h,r n,q are called the n th h, q-Bernoulli numbers of order r.

By2.17, one obtains the following theorem

Theorem 2.4 For h ∈ Z, r ∈ N, one has

β h,r n,q x   1

1− qn

n

l0



n l



−q xl

lh−1

r



lh−1

r



q

r!

r q!,

β h,r n,q x f n−r

q f−1

a1, ,a r0

qr j1 h−ja j β h,r n,q f



x  a1 · · ·  a r

f



.

2.19

Let χ be the primitive Dirichlet’s character with conductor f ∈ N, then one considers the generalized h, q-Bernoulli polynomials attached to χ of order r as follows:

β h,r n,χ,q x 



X

· · ·



X

  

r times

qr j1 h−j−1x j

⎛

⎝r

j1

χ

x j

⎞⎠x  x1 · · ·  x rn

q dμ q x1 · · · dμ q x r .

2.20

By2.20, one sees that

β h,r n,χ,q x f n−r

q f−1

a1, ,a r0

qr j1 h−ja j

⎛

⎝r

j1

χ

a j

⎞⎠β h,r

n,q f



x  a1 · · ·  a r

f



. 2.21

In the special case, x  0, β h,r n,χ,q 0  β n,χ,q h,r are called the n th generalized h, q-Bernoulli numbers attached to χ of order r.

From2.20 and 2.21, one notes that

β h,r n,χ,qq − 1

β n1,χ,q h−1,r  β h−1,r n,χ,q 2.22

By2.16, it is easy to show that

β n,χ,q h,r 



Zp

· · ·



Zp

  

r times

x1 · · ·  x rn

q qr j1 h−j−1x j dμ q x1 · · · dμ q x r





Zp

· · ·



Zp

x1 · · ·  x rn

q



x1 · · ·  x rqq − 1

 1 q

r j1 h−j−2x j

dμ q x1 · · · dμ q x r .

2.23

Thus, one has

β h,r n,q q − 1

β n1,q h−1,r  β h−1,r n,q 2.24

From2.16 and 2.23, one can also derive



Zp

· · ·



Zp

  

r times

q n−2x1n−3x2···n−r−1x r dμ q x1 · · · dμ q x r

Z

p· · ·Z

p q −x1···x rq nx1···x rq −x1−2x2−···−rx r dμ q x1 · · · dμ q x r

n

l0



n

l





q − 1l

Zp· · ·Zp x1 · · ·  x rl

q q −x1···x rq −x1−2x2−···−rx r dμ q x1 · · · dμ q x r

n

l0



n

l





q − 1l

β 0,r l,q ,



Zp

· · ·



Zp

q n−2x1n−3x2···n−r−1x r dμ q x1 · · · dμ q x r 

n−1

r

n−1

r

q

r!

r q!.

2.25

It is easy to see that

n

j0



n

j



q − 1j

Zp

x j

q q h−2x dμ q x 



Zp



q − 1

x q 1 n q h−2x dμ q x

n  h − 1 q .

2.26

By2.23, 2.25, and 2.26, one obtains the following theorem

Theorem 2.5 For h ∈ Z, r ∈ N, and n ∈ Z, one has

β h,r n,q q − 1

β h−1,r n1,q  β h−1,r n,q ,

n

l0



n l





q − 1l

β 0,r l,q 

n−1

r

n−1

r

q

r!

r q!.

2.27

Furthermore, one gets

n

l0



n l



q − 1l

β l,q h,1 n  h − 1

n  h − 1 q . 2.28

Now, one considers the polynomials of β 0,r n,q x by

β 0,r n,q x 



Zp

· · ·



Zp

  

r times

x  x1 · · ·  x rn

q q −2x1−3x2−···−r−1x r dμ q x1 · · · dμ q x r

  1

1− qn

n

l0



n l



−1l

q lx

l−1

r



l−1

r



q

r!

r q!.

2.29

By2.29, one obtains the following theorem

Theorem 2.6 For r ∈ N and n ∈ Z, one has



1− qn

β 0,r n,q x  n

l0



n l



−1l

q lx

l−1

r



l−1

r



q

r!

r q!. 2.30

By using multivariate p-adic q-integral on Z p , one sees that

q nx

n−1

r



n−1

r



q

r!

r q!





Zp

· · ·



Zp

  

r times

q nxn−2x1···n−r−1x r dμ q x1 · · · dμ q x r





Zp

· · ·



Zp



q − 1

x  x1 · · ·  x rq 1 n q −2x1···−r1x r dμ q x1 · · · dμ q x r

 n

l0



n

l



q − 1l

Zp

· · ·



Zp

x  x1 · · ·  x rl

q q −2x1···−r1x r dμ q x1 · · · dμ q x r

 n

l0



n

l



q − 1l

β 0,r l,q x.

2.31

Therefore, one obtains the following corollary

Corollary 2.7 For r ∈ N and n ∈ Z, one has

q nx

n−1

r



n−1

r



q

r!

r q!  n

l0



n l



q − 1l

β l,q 0,r x. 2.32

It is easy to show that



Zp

· · ·



Zp

  

r times

x  x1 · · ·  x rn

q q −2x1···−r1x r dμ q x1 · · · dμ q x r

f n−r

q f−1

i1, ,i r0

q−r l1 li l

×



Zp

· · ·



Zp

q −fr l1 l1x l

x r

l1 i l

l1

x l n

q f

dμ q f x1 · · · dμ q f x r .

2.33

From2.33, one notes that

β 0,r n,q x f n−r

q f−1

i1, ,i r0

q −i1−2i2− −ri r β 0,r n,q f



x  i1 · · ·  i r

f



. 2.34

From the multivariate p-adic q-integral on Z p , one has



Zp

· · ·



Zp

  

r times

x  x1 · · ·  x rn

q q −2x1−3x2−···−r1x r dμ q x1 · · · dμ q x r





Zp

· · ·



Zp



x q  q x x1 · · ·  x rq n q −2x1−3x2−···−r1x r dμ q x1 · · · dμ q x r

 n

l0



n

l



x n−l

q q lx



Zp

· · ·



Zp

x1 · · ·  x rl

q q −2x1−3x2−···−r1x r dμ q x1 · · · dμ q x r ,

2.35



Zp

· · ·



Zp

  

r times

x  y  x1 · · ·  x r

n

q q −2x1−3x2−···−r1x r dμ q x1 · · · dμ q x r

 n

l0



n

l



y n−l

q q ly



Zp

· · ·



Zp

x  x1 · · ·  x rl

q q −2x1−3x2−···−r1x r dμ q x1 · · · dμ q x r .

2.36

By2.35 and 2.36, one obtains the following corollary

Corollary 2.8 For r ∈ N and n ∈ Z, one has

β 0,r n,q x  n

l0



n l



x n−l

q q lx β l,q 0,r ,

β 0,r n,q



x  y

 n

l0



n l



y n−l

q q ly β 0,r l,q x.

2.37

Now, one also considers the polynomial of β h,1 n,q x From the integral equation on Z p , one notes that

β h,1 n,q x 



Zp

x  x1n

q q x1h−2 dμ q x1

  1

1− qn

n

l0



n l



−1l q lx l  h − 1

l  h − 1 q .

2.38

By2.38, one easily gets

β h,1 n,q x   1

1− qn−1

n

l0

n

l−1l

q lx l

1− q lh−1  h − 1

1− qn−1

n

l0

n

l−1l

q lx

1− q lh−1

  −n

1− qn−1

n−1

l0

n−1

l



−1l

q x q lx

1− qn−1

n

l0

n

l−1l

q lx

1− q lh−1

 −n∞

m0

q hmx x  m n−1

q  h − 11− q ∞

m0

q h−1m x  m n

q

2.39

Thus, one obtains the following theorem

Theorem 2.9 For h ∈ Z and n ∈ Z, one has

β h,1 n,q x  −n ∞

m0

q hmx x  m n−1

q  h − 11− q ∞

m0

q h−1m x  m n

q 2.40

From the definition of p-adic q-integral on Z p , one notes that



Zp

q h−2x1x  x1n

q dμ q x1

 1

f

q

f−1

i0

q h−1i i n

q



Z

!

x  i

"n

q f

q fh−2x1dμ q f x1.

2.41

Thus, one has

β h,1 n,q x  1

f

q

f−1

i0

q h−1i i n

q β h,1 n,q f



x  i f



By2.38, one easily gets



Zp

x  x1n

q q x1h−2 dμ q x1

 q −x

Zp

x  x1n

q



x  x1qq − 1

 1 q x1h−3 dμ q x1.

2.43

From2.43, one has

β h,1 n,q x  q −x

q − 1

β n1,q h−1,1 x  β h−1,1 n,q x . 2.44

That is,

q x β n,q h,1 x q − 1

β n1,q h−1,1 x  β h−1,1 n,q x. 2.45

By2.38 and 2.43, one easily sees that



Zp

q h−2x1x  x1n

q dμ q x1 



Zp

q h−2x1



x q  q x x1q n dμ q x1

 n

l0



n l



x n−l

q q lx



Zp

q h−2x1x1l

q dμ q x1,

2.46

and

q h−1



Zp

q h−2x1x1 1  x n

q dμ q x1 −



Zp

q h−2x1x  x1n

q dμ q x1

 q x n x n−1

q  hq − 1

x n

q−q − 1

x n

q

2.47

For x  0, this gives

q h−1



Zp

q h−2x1x1 1n

q dμ q x1 −



Zp

q h−2x1x1n

q dμ q x1 

⎧

⎨

⎩

1, if n  1,

0, if n > 1, 2.48

and

β 0,q h,1



Z q h−2x1dμ q x1  h − 1 h − 1

q

From2.46 and 2.48, one can derive the recurrence relation for β h,1 n,q as follows:

q h−1 β h,1 n,q 1 − β n,q h,1  δ n,1 , 2.50

where δ n,1 is kronecker symbol.

By2.46, 2.48, and 2.50, one obtains the following theorem

Theorem 2.10 For h ∈ Z and n ∈ Z, one has

β h,1 n,q x  n

l0



n l



x n−l

q q lx β h,1 l,q ,

q h−1 β h,1 n,q x  1 − β h,1 n,q  q x n x n−1

q  hq − 1

x n

q−q − 1

x n

q

2.51

Furthermore,

q h−2

q − 1

β h−1,1 n1,q 1  q h−2 β h−1,1 n,q 1 − β h,1 n,q  δ n,1 , 2.52

where δ n,1 is kronecker symbol.

From the definition of p-adic q-integral on Z p , one notes that



Zp

q −h−2x11 − x  x1n

q−1dμ q−1x1

 −1n q nh−2



Zp

q h−2x1x  x1n

q dμ q x1.

2.53

By2.53, one sees that

β h,1 n,q−11 − x  −1 n

q nh−2 β h,1 n,q x. 2.54

Note that

B n 1 − x  lim

q−→1 β h,1 n,q−11 − x  lim

q−→1−1n q nh−2 β h,1 n,q x  −1 n B n x, 2.55

where B n x are the n th ordinary Bernoulli polynomials.

In the special case, x  1, one gets

β h,1 n,q−1  −1n

q nh−2 β h,1 n,q 1  −1n

q n−1 β h,1 n,q , if n > 1. 2.56

Let χ be the Dirichlet’s character with conductor f ∈ N, then the generalized Carlitz’s

q-Bernoulli numbers. .. are called the n th generalized q-Bernoulli polynomials of order r attaches to χ.

By2.13,...

By2.17, one obtains the following theorem