Báo cáo sinh học: "Research Article On the Twisted q-Analogs of the Generalized Euler Numbers and Polynomials of Higher Order" ppt

Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral onZp ,” Russian Journal of Mathematical Physics, vol.. Kim, “Note

Volume 2010, Article ID 875098, 11 pages

doi:10.1155/2010/875098

Research Article

Numbers and Polynomials of Higher Order

1 Department of Mathematics and Computer Science, KonKuk University,

Chungju 138-701, Republic of Korea

2 Department of Wireless Communications Engineering, Kwangwoon University,

Seoul 139-701, Republic of Korea

3 Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Correspondence should be addressed to Lee Chae Jang,leechae.jang@kku.ac.kr

Received 12 April 2010; Accepted 28 June 2010

Academic Editor: Istvan Gyori

Copyrightq 2010 Lee Chae Jang 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 consider the twisted q-extensions of the generalized Euler numbers and polynomials attached

to χ.

1 Introduction and Preliminaries

Let p be an odd prime number For n ∈ Z  N ∪ {0}, let C p n  {ζ | ζ p n

 1} be the

cyclic group of order p n , and let T p  limn → ∞ C p n  n≥0 C p n  C p∞ be the space of locally

constant functions in the p-adic number field C p When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ C p If q ∈ C,

one normally assumes that|q| < 1 If q ∈ C p, one normally assumes that|1 − q| p < 1 In this

paper, we use the notation

x q 1− q x

1− q , x −q 1−



−qx

Let d be a fixed positive odd integer For N ∈ N, we set

X  X d  lim←−NZ

dpNZ, X1 Zp ,

X∗ 

0<a<dp

a,p1



a  dpZ p



,

a  dp nZpx ∈ X | x ≡ a

mod dp n

,

1.2

where a ∈ Z lies in 0 ≤ a < dp n; compared to1 16

Let χ be the Dirichlet’s character with an odd conductor d ∈ N Then the generalized

ζ-Euler polynomials attached to χ, E n,χ,ζ x, are defined as

F χ,ζ x, t  2

d−1

l0 −1l χ lζ l e lt

ζ d e dt 1 e xt

 ∞

n0

E n,χ,ζ x t n

n! , for ζ ∈ T p .

1.3

In the special case x  0, E n,χ,ζ  E n,χ,ζ 0 are called the nth ζ-Euler numbers attached to χ For f ∈ UDZ p , the p-adic fermionic integral on Z pis defined by

I−q

f



Zp

f xdμ −q x  lim

N → ∞

p N−1

x0

f xμ −q

x  p NZp

 lim

N → ∞

p N−1

x0

f x−1 x q x

p N

−q , see 7–17

1.4

Let I−1 limq → 1  I −q f Then, we see that

Zp

f xdμ−1x 

X

For n ∈ N, let f n x  fx  n Then, we have

Zp

f x  ndμ−1x  −1 n

Zp

f xdμ−1x  2 n−1

l0

−1n−1−l f l. 1.6

Thus, we have

I−1

f n



 −1n−1 I−1

f

 2n−1

l0

−1n−1−l f l, see 7–17. 1.7

By1.7, we see that

X

χ

y

ζ y e xyt dμ−1

y

 2

d−1

l0 −1l

χ lζ l e lt

ζ d e dt 1 e xt

∞

n0

E n,χ,ζ x t n

From1.8, we can derive the Witt’s formula for E n,χ,ζ x as follows:

X

χ xx n ζ x dμ−1 x  E n,χ,ζ ,

X

χ

y

y  xn

ζ y dμ−1

y

 E n,χ,ζ x, for ζ ∈ T p , see 5–17.

1.9

The nth generalized ζ-Euler polynomials of order k, E k n,χ,ζ, are defined as



2d−1

l0 ζ l−1l χle lt

ζ d e dt 1 e xt

k

 ∞

n0

E k n,χ,ζ x t n

In the special case x  0, E k n,χ,ζ  E k n,χ,ζ 0 are called the nth ζ-Euler numbers of order k attached to χ.

Now, we consider the multivariate p-adic invariant integral on X as follows:

X

· · ·

X

 k



i1

χ x i



e x1···x k xt ζ x1···x k dμ−1 x1 · · · dμ−1x k





2d−1

l0 −1l

χle lt

ζ d e dt 1

k

e xt ∞

n0

E k n,χ,ζ x t n

n! .

1.11

By1.10 and 1.11, we see the Witt’s formula for E k n,χ,ζ x as follows:

X

· · ·

X

k

i1

χ x i



x1 · · ·  x k  x n ζ x1···x k dμ−1 x1 · · · dμ−1x k   E k n,χ,ζ x. 1.12

The purpose of this paper is to present a systemic study of some formulas of the

twisted q-extension of the generalized Euler numbers and polynomials of order k attached

to χ.

In this section, we assume that q ∈ C pwith|1 − q| p < 1 and ζ ∈ T p For d ∈ N with 2  d, let

χ be the Dirichlet’s character with conductor d For h ∈ Z, k ∈ N, let us consider the twisted

h, q-extension of the generalized Euler numbers and polynomials of order k attached to χ.

We firstly consider the twisted q-extension of the generalized Euler polynomials of higher

order as follows:

∞

n0

E n,χ,ζ,q x t n

n! 

X

e xy q t ζ y χ

y

dμ−1

y

 2 ∞

m0

χ m−1 m ζ m e mx q t

2.1

By2.1, we see that

X

x  yn

q χ

y

ζ y dμ−1

y

 2 ∞

m0

χ m−ζ m

e mx q t

 2d−1

a0

χ a−1 a

ζ a 1

1− qn

n

l0



n l



−1l ζ la q lax

1 q ld ζ ld

2.2

From the multivariate fermionic p-adic invariant integral on Z p, we can derive the twisted

q-extension of the generalized Euler polynomials of order k attached to χ as follows:

∞

n0

E k n,χ,ζ,q x t n

n! 

X

· · ·

X

 k



i1

χ x i



e x1···x k x q t

ζ x1···x k dμ−1 x1 · · · dμ−1x k . 2.3

Thus, we have

E n,χ,ζ,q k x 

X

· · ·

X

 k



i1

χ x i



x1 · · ·  x k  x n

q ζ x1···x k dμ−1 x1 · · · dμ−1x k

 d−1

a1, ,a k0

 k



i1

χ a i



−ζk j1 a j 2k



1− qn

n

l0

n

l−1l

q lxk j1 a j



1 q ld ζ d

 2k d−1

a1, ,a k0

k



i1



χ a i−ζk j1 a j

∞

m0



m  k − 1 m



×−ζ d m

x  a1 · · ·  a k  md n

q

2.4

Let F q,χ,ζ k t, x  ∞n0 E k n,χ,ζ,q xt n /n! be the generating function for E n,χ,ζ,q k x By 2.3,

we easily see that

F k q,χ, t, x  2 k d−1

a1, ,a k0

 k



i1

χ a i



−ζk j1 a j

∞

m0



m  k − 1 m



× −ζ dm

e xa1···a k md q t

 2k ∞

n1, ,n k0

−ζ n1···n k

k

i1

χ n i



e n1···n k x q t

2.5

Therefore, we obtain the following theorem

Theorem 2.1 For k ∈ N, n ≥ 0, one has

E k n,χ,ζ,q x  2 k ∞

n1, ,n k0

−ζ n1···n k

k

i1

χ n i



n1 · · ·  n k  x n

q

 d−1

a1, ,a r0

 k



i1

χ a i



−ζk j1 a j 2k



1− qn

n

l0

n

l−1l

q lxk j1 a j



1 q ld ζ dn

2.6

Let h ∈ Z, r ∈ N Then we define the extension of E r n,χ,ζ,q x as follows:

∞

n0

E h,r n,χ,ζ,q x t n

n! 

X

· · ·

X

qr j1 h−jx j

 k



i1

χ x i



e xr j1 x jq t × ζ x1···x r dμ−1x1 · · · dμ−1x r .

2.7

Then, E r n,χ,ζ,q x are called the nth generalized h, q-Euler polynomials of order r attached

to χ In the special case x  0, E r n,χ,ζ,q  E r n,χ,ζ,q 0 are called the nth generalized h, r-Euler numbers of order r By 1.7, we obtain the Witt’s formula for E r n,χ,ζ,q x as follows:

E h,r n,χ,ζ,q x 

X

· · ·

X

qr j1 h−jx j

k

i1

χ x i

⎡

⎣x  r

j1

x j

⎤

⎦

n

q

ζ x1···x r dμ−1x1 · · · dμ−1x r

 d−1

a1, ,a r0

r

i1

χ a i



−ζr j1 a j qr j1 a j h−j× 2r

1− qn

n

l0

n

l−1l q lxr j1 a j



−q dh−rl ζ d ; q d

r

,

2.8 wherea; q r  1 − a1 − aq · · · 1 − aq r−1

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

k q k − 1 q· · · 2q1q From2.8, we note that

E n,χ,ζ,q h,r x   2r

1− qn

d−1

a1, ,a r0

 r



i1

χ a i



−ζr j1 a j qr j1 h−ja j

× n

l0



n l



−1l q lxr j1 a j ∞

m0



m  r − 1 m



q d

−ζ d m

q dh−rm q ldm

 2r d−1

a1, ,a r0

r−1

i1

χ a i



−ζr j1 a j qr j1 h−ja j

× ∞

m0



m  r − 1 m



q d

−ζ d m

q dh−rm 1

1− qn

1− q dmxk

j1 a j /d n

 2r d n

q

∞

m0



m  r − 1 m



q d

−ζ d m

q dh−rm

d−1

a1, ,a r0

r−1



i1

χ a i



× −ζr j1 a j qr j1 h−ja j

⎡

⎣m  x 

d−1

j1 a j

d

⎤

⎦

n

q d

.

2.9

Let F q,χ,ζ h,r t, x ∞n0 E n,χ,ζ,q h,r xt n /n! be the generating function for E h,r n,χ,ζ,q x From

2.8, we can easily derive

F q,χ,ζ h,r t, x  2 r ∞

n1, ,n r0

qr j1 h−jn j −ζr j1 n j

⎛

⎝r

j1

χ

n j

⎞⎠e n1···n r x q t

 2r ∞

m0



m  r − 1 m



q

−ζ d m

q dh−rm

d−1

a1, ,a r0

r−1

i1

χ a i



× −ζr j1 a j qr j1 h−ja j e mdxr j1 a jq t

.

2.10

By2.10, we obtain the following theorem

Theorem 2.2 For h ∈ Z, r ∈ N, one has

E n,χ,ζ,q h,r x  2 r ∞

n1, ,n r0

qr j1 h−jn j −ζr j1 n j

⎛

⎝r

j1

χ

n j

⎞⎠n1 · · ·  n r  x n

q

 2r d n

q

∞

m0



m  r − 1 m



q

−ζ d m

q dh−rm

d−1

a1, ,a r0

r−1



i1

χ a i



× −ζr j1 a j qr j1 h−ja j

m  x 

r

j1 a j

d

n

q d

 d−1

a1···a r0

⎛

⎝r

j1

χ

n j

⎞⎠−ζr

j1 a j qr j1 h−ja j× 2r

1− qn

n

l0

n

l−1l q lxr j1 a j



−q dh−rl ζ d ; q d

r

.

2.11

Let h  r Then we see that

E r,r n,χ,ζ,q x   2r

1− qn

d−1

a1, ,a r0

r−1

i1

χ a i



−ζr i1 a i qr j1 h−ja j× n

l0

n

l−1l q lr j1 a j x



−q ld ζ d ; q d

r

 2r d n

q

∞

m0



m  r − 1 m



q

−ζ m d−1

a1, ,a r0

 r



i1

χ a i



× −ζr j1 a j qr j1 r−ja j

m  x 

r

j1 a j

d

n

q d

.

2.12

It is easy to see that

X

· · ·

X

 k



i1

χ x i



qr j1 h−jx j xm ζ x1···x r dμ−1 x1 · · · dμ−1x r

 d−1

a1, ,a r0

 r



i1

χ a i



q mxr j1 h−ja j −ζr j1 a j×

X

· · ·

X

qr j1 m−jx j dμ−1 x1 · · · dμ−1x r

 2

r q mxd−1

a1, ,a r0r

j1 χ

a j



qr j1 m−ja j −ζr j1 a j



−q dm−r ζ d ; q d

r

.

2.13

Thus, we have

2r q mxd−1

a1, ,a r0r

j1 χ

a j



qr j1 m−ja j −ζr j1 a j



−q dm−r ζ d ; q d

r



X

· · ·

X

x  x1 · · ·  x rqq − 1

 1m q−r j1 jx j ζ x1···x r

×

⎛

⎝r

j1

χ

x j

⎞⎠dμ−1x1 · · · dμ−1x r

 m

l0



m l



q − 1l

X

· · ·

X

⎛

⎝r

j1

χ

x j

⎞⎠

× x  x1 · · ·  x rl

q q−r j1 jx j ζ x1···x r dμ−1 x1 · · · dμ−1x r

 m

l0



m l



q − 1l

E 0,r l,χ,ζ,q x.

2.14

By2.14, we obtain the following theorem

Theorem 2.3 For d, k ∈ N with 2  d, one has

2r q mxd−1

a1, ,a r0r

j1 χ

a j



qr j1 m−ja j −ζr j1 a j



−q dm−r ζ d ; q d

r

 m

l0



m l



q − 1l

E l,χ,ζ,q 0,r x. 2.15

By1.7, we easily see that

X

f x  ddμ−1x 

X

f xdμ−1x  2 d−1

l0

−1l f l. 2.16 Thus,we have

q dh−1

X

· · ·

X

x  d  x1 · · ·  x rn

q qr j1 r−jx j ζr j1 x j

×

⎛

⎝r

j1

χ

x j

⎞⎠dμ−1x1 · · · dμ−1x r

 −

X

· · ·

X

x  x1 · · ·  x rn

q qr j1 r−jx j ζr j1 x j×

⎛

⎝r

j1

χ

x j

⎞⎠dμ−1x1 · · · dμ−1x r

 2d−1

l0

χ l−ζ l

X

· · ·

X

x  l  x2 · · ·  x rn

q

⎛

⎝r−1

j1

χ

x j1

⎞⎠

× qr−1 j1 x j1 h−1−j ζ x2x3···x r dμ−1x2 · · · dμ−1x r .

2.17

By2.17, we obtain the following theorem

Theorem 2.4 For h ∈ Z, d ∈ N with 2  d, one has

q dh−1 E h,r n,χ,ζ,q x  d  E h,r n,χ,ζ,q x  2 d−1

l0

χ l−1 l E h−1,r−1 n,χ,ζ,q x  l. 2.18

It is easy to see that

q x E h1,r n,χ,ζ,q x q − 1

E h,r n1,χ,ζ,q  E n,χ,ζ,q h,r x. 2.19

Let F q,χ,ζ h,1 t, x ∞n0 E h,1 n,χ,ζ,q xt n /n! Then we note that

F q,χ,ζ h,1 t, x  2 ∞

n0

χ nq h−1n −ζ n

e nx q t 2.20

From2.20, we can derive

E h,1 n,χ,ζ,q x  2 ∞

m0

χ mq h−1m −ζ m m  x n

q   2

1− qn

d−1

a0

χ a−ζ a n

l0

n

l−1l q lxa



1 q ld ζ d  2.21

3 Further Remark

In this section, we assume that q ∈ C with |q| < 1 Let χ be the Dirichlet’s character with an odd conductor d ∈ N From the Mellin transformation of F h,r q,χ,ζ t, x in 2.10, we note that

1

Γs

F q,χ,ζ h,r −t, xt s−1 dt  2 r

∞

m1, ,m r0

qr j1 h−jm j −ζ m1···m rr

j1 χ

m j



m1 · · ·  m r  x s

q

, 3.1

where h, s ∈ C, x /  0, −1, −2, , and r ∈ N, ζ  e 2πi/d By3.1, we can define the Dirichlet’s type multipleh, q-l-function as follows.

Definition 3.1 For s ∈ C, x ∈ R with x /  0, −1, −2, , one defines the Dirichlet’s type multiple

h, q-l-function related to higher order h, q-Euler polynomials as

l h,r q



s, x | χ

 2r ∞

m1,··· ,m r0

qr j1 h−jm j −ζ m1···m rr

i1 χ m i

m1 · · ·  m r  x s

q

where s, h ∈ C, x /  0, −1, −2, · · · , r ∈ N, and ζ  e 2πi/d

Note that l h,r q s, x | χ is analytic continuation in whole complex s-plane In 2.10, we note that

F q,χ,ζ h,r t, x  2 r ∞

n1, ,n r0

qr j1 h−jn j −ζ n1···n r

⎛

⎝r

j1

χ

n j

⎞⎠e n1···n r x q t

 ∞

n0

E h,r n,χ,ζ,q x t n

n! .

3.3

By Laurent series and Cauchy residue theorem in 3.1 and 3.3, we obtain the following theorem

Theorem 3.2 Let χ be Dirichlet’s character with odd conductor d ∈ N, and let ζ  e 2πi/d For

h, s ∈ C, x /  0, −1, −2, , r ∈ N, and n ∈ Z, one has

l h,r q



−h, x | χ E n,χ,ζ,q h,r x. 3.4

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The purpose of this paper is to present a systemic study of some formulas of the

twisted q-extension of the generalized Euler numbers and polynomials of order k attached...

17 Y.-H Kim, W Kim, and C S Ryoo, ? ?On the twisted q -Euler zeta function associated with twisted< /i>

q -Euler numbers, ” Proceedings of the Jangjeon Mathematical Society, vol 12,... class="text_page_counter">Trang 4

We firstly consider the twisted q-extension of the generalized Euler polynomials of higher< /i>

order as follows:

∞