Some Identities on the

Volume 2010, Article ID 860280,14pages doi:10.1155/2010/860280

Research Article

Some Identities on the

q

-Genocchi Polynomials

of Higher-Order and

q

-Stirling Numbers by the

Fermionic

p

-Adic Integral on

Z

p

Seog-Hoon Rim, Jeong-Hee Jin, Eun-Jung Moon,

and Sun-Jung Lee

Department of Mathematics, Kyungpook National University, Tagegu 702-701, Republic of Korea

Correspondence should be addressed to Seog-Hoon Rim,[email protected]

Received 25 September 2010; Accepted 8 November 2010

Academic Editor: H. Srivastava

Copyrightq2010 Seog-Hoon Rim 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.

A systemic study of some families ofq-Genocchi numbers and families of polynomials of N ¨orlund type is presented by using the multivariate fermionicp-adic integral onZp. The study of these higher-orderq-Genocchi numbers and polynomials yields an interestingq-analog of identities for Stirling numbers.

1. Introduction

Letpbe a fixed odd prime number. Throughout this paper,Zp,Qp,C, andCpdenote the ring

ofp-adic rational integers, the field ofp-adic rational numbers, the complex number field, and the completion of the algebraic closure ofQp, respectively. LetNbe the set of natural

numbers andZ N∪ {0}. Letvpbe the normalized exponential valuation ofCpwith|p|_p

p−vpp_1/p.

When one talks of q-extension, q is variously considered as an indeterminate, a complexq ∈ C, or ap-adic numberq ∈ Cp. Ifq ∈ C, then one normally assumes|q| < 1.

Ifq∈Cp, then we assume|q−1|_p<1. In this paper, we use the following notation:

xq

1−qx

1−q, x−q

1−−qx

1q , 1.1

Theq-factorial is defined asnq! nqn−1q· · ·2q1q, and the Gaussian binomial

coefficient is defined by the standard rule

n k

q

nq! n−kq!kq!

nqn−1q· · ·n−k1q! kq!

, 1.2

see7,9. Note that limq→1nkq nk n!/n−k!k!nn−1· · ·n−k1/k!. It readily

follows from1.2that

n1

k

q

n k−1

q

qk

n k

q

qn−k1

n k−1

q

n k

q

, 1.3

see4,7.

Theq-binomial formulas are known,

b;q_n 1−b1−bq· · ·1−bqn−1

n

i0

n

i

q

q

i

2−1ibi,

1

b;q_n

1

1−b1−bq· · ·1−bqn−1

∞

i0

ni−1

i

q

bi.

1.4

We say thatf :Zp → Cp is uniformly differentiable function at a pointa ∈Zp, and

we writef ∈ UDZp, if the difference quotientsΦf : Zp×Zp → Cp such thatΦfx, y

fx−fy/x−yhave a limitfaasx, y → a, a. Forf∈UDZp, theq-deformed

fermionicp-adic integral is defined as

I−q

f

Zp

fxdμ−qx lim N→ ∞

1

pN

−q pN₋₁

x0

fx−qx, 1.5

see7,9. Note that

I−1

flim

q→1I−q

f

Zp

fxdμ−1x. 1.6

Forn∈N, writefnx fxn. Then, we have

I−1

fn

−1nI−1

f2

n−1

l0

Using1.7, we can readily derive the Genocchi polynomials,Gnx, namely,

t

Zp

exytdμ−1

y 2t et₁e

xt∞

n0

Gnxt n

n!, 1.8

see1–27. Note thatGn0 Gn are referred to as thenth Genocchi numbers. Let us now

introduce the Genocchi polynomials of N ¨orlund type as follows:

tr

Zp

· · ·

Zp

rtimes

exx1···xrt_dμ−

1x1· · ·dμ−1xr

2t et₁

r

ext ∞

n0

Gnrxt n

n!, 1.9

et₁

2t r

ext ∞

n0

G−nrxt n

n!, 1.10

see7,9. In the special casex0, G−nr0 G−nr, andGnr0 Gnrare referred to as the

Genocchi numbers of N ¨orlund type. LetEhx hx1be the shift operator. Then, the

q-difference operatorΔqis defined as

Δn q

n

i1

E−qi−1I, whereIhx hx, 1.11

see4,7,9. It follows from1.11that

fx

n≥0

x n

q

Δn

qf0, 1.12

whereΔn qf0

_n

k0nkqq k

2_fn−_k see5,6,10. Theq-Stirling number of the second kindas defined by Carlitzis given by

S2

n, k;q q −k

2

kq! k

j0

−1jqj2

k j

q

k−jn_q, 1.13

see7,10. By1.12and1.13, we see that

S2

n, k;q q −k

2

k_q!Δ

k

q0n, 1.14

see6,10.

numbers of N ¨orlund type. The purpose of this paper is to present a systemic study of some families q-Genocchi numbers and polynomials of N ¨orlund type by using the multivariate fermionicp-adic integral onZp.

2.

q

-Extensions of Genocchi Numbers and

Polynomials of N ¨orlund Type

In this section, we assume thatq∈Cpwith|1−q|p<1. We first consider theq-extensions of

1.8given by the rule

∞

n0

Gn,qxt n

n! t _Zpe

xyqt_dμ− 1

y

∞

n0 2t

1−qn

n

l0

n

l−1lqlx

1ql

tn

n! 2t

∞

m0

−1memxqt_.

2.1

Thus, we obtain the following lemma.

Lemma 2.1. Ifn≥0, then

Gn1,qx

n1 2

∞

m0

−1mmxnq

2

1−qn

n

l0

n

l−1lqlx

1ql . 2.2

By1.14,

xnq n

k0

x k

q

kq!S2

k, n−k;qqk2

n

k0

xqx−1q· · ·x−k1q

qk2−n−2k

n−k_q!Δ

n−k q 0k

n

k0

qk2−n−2k

n−kq!

Δn−k q 0k

1

1−qk

k

l0

k

l

q

q2l−₁l_qlx−k1_.

2.3

Thus, we have

Gn1,q

n1

n

k0

qk2_{S2k, n}−_k;q

1−qk

k

l0

k

l

q

q2l−₁l

l

m0

l m

q−1mGm1,q1−k

m1 , 2.4

Theorem 2.2. Ifn≥0, then

Gn1,q

n1

n

k0

qk2_{S2k, n}−_k;q

1−qk

k

l0

k

l

q

q2l−₁l

l

m0

l m

q−1mGm1,q1−k

m1 , 2.5

whereGn,q Gn,q0stand for the nth Genocchi numbers.

Consider aq-extensoin of1.9such thatG₀r_,qx G₁r_,qx · · ·G_rr_−1,qx 0 and

Gnrr,qx

r!nr r

Zp

· · ·

Zp

xx1· · ·xrnqdμ−1x1· · ·dμ−1xr

2r

1−qn

n

l0

n

l

−1lqlx

1 1ql

r

2r

∞

m0

mr−1

m

−1mmxnq.

2.6

LetFqrt, x

_∞

n0G

r

n,qxtn/n!. Then,

Fqrt, x 2rtr

∞

m0

mr−1

m

−1memxqt_{. 2.7}

In the special case x 0, the numbers Gn,qr0 Gn,qr are referred to as q-extension of

the Genocchi numbers of orderr. In the sense of theq-extension in1.10, consider the q -extension of Genocchi polynomials of N ¨orlund type given by

Gqrt, x Fq−rt, x

1 2r_tr

r

m0

r m

emxqt

∞

n0

G−n,qrxt n

n!. 2.8

By2.8,G−_0,qrx G−_1,qrx · · ·G_r−₋r_1,qx 0 andr!nrGnr−r,qx 1/2r

r

m0mrmxnq.

Therefore, we obtain the following theorem.

Theorem 2.3. Forr∈N, and,n≥0, write

2r_tr∞ m0

mr−1

m

−1memxqt

∞

n0

Gn,qrxt n

n!. 2.9

Then,

Gnrr,qx

r!n_rr 2r

1−qn

n

l0

n

l

−1lqlx

1 1ql

r

2r

∞

m0

mr−1

m

−1mmxn_q,

r!

n r

G−n−rr,q x

1 2r_1−qn

n

l0

n

l

−1lqlx1qlr 1

2r r

m0

r m

mxnq.

The numbersG−n,qr0 G−n,qr are referred to as theq-extension of Genocchi numbers

of N ¨orlund type. For h ∈ Z and r ∈ N, introduce the extended higher-order q-Genocchi polynomials as follows:

Gnh,rr,qx

r!nrr

Zp

· · ·

Zp

qrj1h−jxj_xx

1· · ·xrnqdμ−1x1· · ·dμ−1xr. 2.11

Then,

Gnh,rr,qx

r!nr r

2r

1−qn

n

l0

n

l−1lqlx

−qh−1l_;q−1

r

2r

1−qn

n

l0

n

l−1lqlx

−qh−rl_;q r

2r

∞

m0

mr−1

m

q

−1mqh−rmxmnq.

2.12

LetFqh,rt, x

_∞

n0Gn,qh,rxtn/n!. Then, we can readily see that

Fqh,rt, x 2rtr

∞

m0

mr−1

m

q

−1mqh−rmexmqt_{. 2.13}

Therefore, we obtain the following theorem.

Theorem 2.4. Forh∈Zandn≥0, let

2rtr ∞

m0

mr−1

m

q

−1mqh−rmexmqt

∞

n0

Gn,qh,rxt n

n!. 2.14

Then,

Gnh,rr,qx

r!nr r

2r

1−qn

n

l0

n

l−1lqlx

−qh−rl_;q r

2r

∞

m0

mr−1

m

q

−1mqh−rmxmnq. 2.15

Let us now define the extended higher-order N ¨orlund typeq-Genocchi polynomials as follows:

r!

n r

Gnh,−r,q−rx

1

1−qn

n

l0

n

l−1lqlx

Zp· · ·

Zpq

lx1···xr_q r

j1h−jxj_dμ−

1x1· · ·dμ−1xr

By2.16,

r!

n r

Gnh,−−r,qrx

1 2r_1−qn

n

l0

n

l

−1lqlx−qh−rl;q

r

1

2r r

m0

r m

q

qm2_qh−rm_mxn_q.

2.17

LetFqh,−rt, x ∞n0G

h,−r

n,q xtn/n!. Then, we have

Fqh,−rt, x

1 2r_tr

r

m0

r m

q

qm2_qh−rm_emxqt_{, 2.18}

where, G₀h,_,q−rx G₁h,_,q−rx · · · G_rh,₋₁−_,qrx 0. Therefore, we obtain the following theorem.

Theorem 2.5. Forh∈Z,n≥0, andr ∈N, write

1 2r_tr

r

m0

r m

q

qm2_qh−rm_emxqt

∞

n0

Gn,qh,−rxt n

n!. 2.19

Then,

r!

n r

Gnh,−−r,qrx

1 2r_1−qn

n

l0

n

l

−1lqlx_−qh−rl_;q r

1

2r r

m0

r m

q

qm2_qh−rm_mxn_q,

2.20

where,G₀h,_,q−rx G₁h,_,q−rx · · ·G_rh,₋₁−_,qrx 0. Forhr,

Gnr,rr,qx

r!nr

r

2r

1−qn

n

l0

_n

l

−1lqlx

−ql_;q r

2r

∞

m0

mr−1

m

q

−1mxmnq, 2.21

r!

n r

Gnr,−−r,qrx

1 2r_1−qn

n

l0

n

l

−1lqlx−ql;q

r

1 2r

r

m0

r m

q

q

m

2_mxn_q.

It can readily be seen that

qmx₂r

−qm−r_;q r

Zp

· · ·

Zp

qrj1m−jxjmx_dμ−

1x1· · ·dμ−1xr

Zp

· · ·

Zp

xx1· · ·xrq

q−11mq−rj1jxj_dμ

−1x1· · ·dμ−1xr

m

l0

m

l

q−1l

Zp

· · ·

Zp

xx1· · ·xrlqq−

r j1jxj_dμ

−1x1· · ·dμ−1xr

m

l0

m

l

q−1lG

0,r lr,qx

r!lr r

.

2.23

By2.23,qmx₂r_/−qm−r_;q r

m

l0ml q−1lG

0,r

lr,qx/r!

_lr

r

. As is known,

I−1

f1

I−1

f2f0, where f1x fx1. 2.24

It follows from2.24that

qh−1

Zp

· · ·

Zp

x1x1· · ·xrnqq

r

j1h−jxj_dμ

−1x1· · ·dμ−1xr

−

Zp

· · ·

Zp

xx1· · ·xrnqq

r

j1h−jxj_dμ−

1x1· · ·dμ−1xr

2

Zp

· · ·

Zp

xx2· · ·xrnqq

r−1

j1h−1−jxj1_dμ−

1x2· · ·dμ−1xr.

2.25

By2.25,

qh−1G h,r nr,qx1

nr

Gnh,rr,qx

nr 2G h−1,r−1

nr−1,q x. 2.26

A simple manipulation shows that

qx

Zp

· · ·

Zp

xx1· · ·xrnqq

r

j1h−j1xj_dμ−

1x1· · ·dμ−1xr

q−1

Zp

· · ·

Zp

xx1· · ·xrnq1q

r

j1h−jxj_dμ−

1x1· · ·dμ−1xr

Zp

· · ·

Zp

xx1· · ·xrnqq

r

j1h−jxj_dμ−

1x1· · ·dμ−1xr.

2.27

By2.27,qx_Gh1,r

nr,q x/n1 q−1Gnh,rr1,qx/nr1 G

h,r

Therefore, we obtain the following proposition.

Proposition 2.6. Forh∈Z,r ∈Nandn≥0, the following equations

qh−1G h,r

nr,qx1

nr

Gnh,rr,qx

nr 2G h−1,r−1

nr−1,q x,

qxG h1,r nr,q x

n1

q−1G

h,r

nr1,qx

nr1

Gnh,rr,qx

n1

2.28

hold. Moreover,qmx2r/−qm−r;q_r ml0mlq−1lGl0,rr,qx/r!

_lr

r

.

By2.21,

G_nr,r_r,q−1r−x

r!nr r

2r

1−q−1n

n

l0

n

l−1lq−lr−x

−q−l_;q−1

r

−1nqnr2 2

r

1−qn

n

l0

n

l−1lqx

−ql_;q r

−1nqnr2G

r,r

nr,qx

r!nrr

.

2.29

Hence,

Zp

· · ·

Zp

r−xx1· · ·xrn_q−1q−

r

j1r−jxj_dμ

−1x1· · ·dμ−1xr

−1nqnr2

Zp

· · ·

Zp

xx1· · ·xrnqq

r

j1r−jxj_dμ−

1x1· · ·dμ−1xr.

2.30

Forhr,G_nr,r_r,q−10 −1nqn

r

2_Gnr,r_r,qr. It also follows from2.26that

qr−1G r,r

nr,qx1

nr

Gnr,rr,q x

nr 2G r−1,r−1

nr−1,q x. 2.31

The Stirling numbers of the first kind are defined as

n

k1

1 kqz

n

k0

S1

n, k;qzk, 2.32

see6,9,

qm2

r m

q

q m_2r

q· · ·r−m1q

mq!

1

mq! m−1

k0

It can readily be seen that

n−1

k0

z−kq

zn

n−1

k0

1− kq

z

n

k0

S1

n−1, k;q−1kzn−k. 2.34

By2.33and2.34,

m−1

k0

rq−kq

m

k0

S1

m−1, k;q−1krmq−k. 2.35

Formulas2.22and2.35imply the following assertion.

Proposition 2.7. Forr∈Nandn∈Z,

r!

n r

Gnr,−−r,qrx

1 2r_m

q! r

m0

m

k0

S1

m−1, k;q−1krqm−kmxnq. 2.36

The generalized Genocchi numbers and polynomials of N ¨orlund type are defined by

2r_tr

ew1t1ew2t1· · ·ewrt1e

xt∞

n0

Gnrx|w1, . . . , wrt n

n!, 2.37

andGnrw1, . . . , wr Gnr0|w1, . . . , wr. We can now also define aq-extension of2.37as

follows. Forw1, . . . , wr ∈Zpandδ1, . . . , δr ∈Z, write

Gnrr,qx|w1, . . . , wr;δ1, . . . , δr

r!nrr

Zp

· · ·

Zp

x1w1· · ·xrwrxnqdμ−qδ1x1· · ·dμ−qδrxr,

2.38

andGnrr,qw1, . . . , wr;δ1, . . . , δr Gnrr,q0|w1, . . . , wr;δ1, . . . , δr. Thus,

Gnrr,qx|w1, . . . , wr;δ1, . . . , δr

r!nrr

2qδ1· · ·2qδr

1−qn

n

l0

n

l−1lqlx

1qδ1lw1· · ·1qδrlwr. 2.39

Anotherq-extension of N ¨orlund type generalized Genocchi numbers and polynomials is also of interest, namely,

G∗nrr,q x|w1, . . . , wr;δ1, . . . , δr

r!nr r

Zp

· · ·

Zp

x1w1· · ·xrwrxnqqδ1x1···δrxrdμ−1x1· · ·dμ−1xr,

andG∗nrr,q w1, . . . , wr;δ1, . . . , δr G∗nrr,q 0|w1, . . . , wr;δ1, . . . , δr. By2.40,

G∗nrr,q x|w1, . . . , wr;δ1, . . . , δr

r!nr r

2r

1−qn

n

l0

n

l−1lqlx

1qδ1lw1· · ·1qδrlwr. 2.41

3. Further Remarks

Forh0, consider the following polynomialsGn0,rr,qx/r!nrrandr!nrGn0,−r,qrx:

Gn0,rr,qx

r!nr

r

Zp

· · ·

Zp

xx1· · ·xrnqq−

r j1jxj_dμ−

1x1· · ·dμ−1xr,

r!

n r

Gn0−,−r,qrx

1

1−qn

n

l0

_n

l

−1lqlx

Zp· · ·

Zpq

lx1···xr_q− r

j1jxj_dμ−

1x1· · ·dμ−1xr

.

3.1

Then,

Gn0,rr,qx

r!n_rr 2r

1−qn

n

l0

_n

l

−1lqlx

−ql−r_;q r

2r

∞

m0

mr−1

m

q

q−rm−1mxmnq

r!

n r

Gn0−,−r,qrx

1 2r_1−qn

n

l0

n

l

−1lqlx−ql−r;q

r

1 2r

r

m0

r m

q

q

m

2_q−rm_mxn_q.

3.2

LetFq0,rt, x

_∞

n0Gn,q0,rxtn/n!and letFq0,−rt, x

_∞

n0Gn,q0,−rxtn/n!. Then,

Fq0,rt, x 2rtr

∞

m0

mr−1

m

q

q−rm−1mexmqt_,

Fq0,−rt, x

1 2r_tr

r

m0

r m

q

q

m

2_q−rm_emxqt_.

3.3

Consider the following polynomials:

G_nh,₁1_,qx

n1 _Zpq

x1h−1_xx

1nqdμ−1x1

2

1−qn

n

l0

n

l−1lqlx

A simple calculation of the fermionicp-adic invariant integral onZpshow that

qx Zp

xx1nqqx1h−1dμ−1x1

q−1

Zp

xx1nq1qx1h−2dμ−1x1

Zp

xx1nqqx1h−2dμ−1x1.

3.5

By3.5,qx_Gh,1

n1,qx q−1G

h−1,1

n2,q x/2n2 G

h−1,1

n1,q x. It can readily be proved that

Zp

xx1nqqx1h−1dμ−1x1

n

j0

n j

xnq−jqjx Zp

x1jqqx1h−1dμ−1x1. 3.6

By3.6,G_nh,1_1,qx/n1 n_j0nj

xnq−jqjxG_jh,₁1_,q/j1. Using2.24, we can also prove

that

Zp

xx11nqqx11h−1dμ−1x1

Zp

xx1nqqx1h−1dμ−1x1 2xnq. 3.7

Thus,qh−1_Gh,1

n1,qx/n1 G

h,1

n1,qx/n1 2x n

q. Forx0, we haveqh−1G h,1

n1,q1/

n1 G_nh,₁1_,q/n1 2δn,0, whereδn,0is the Kronecker delta. It is easy to see thatG₁h,_,q1_Zpqx1h−1_dμ−

1x1 2/1qh−1 2/2qh−1. By3.4,

G_nh,1_1,q−11−x

n1 _Zp1−xx1

n

q−1q−x1h−1dμ−1x1

−1nqnh−1 2

1−qn

n

l0

n

l−1lqlx

1qlh−1

−1nqnh−1G h,1

n1,qx

n1 .

3.8

In particular, if x 1, then G_nh,₁1_,q−10/n 1 −1nqnh−1G_nh,₁1_,q1/n 1

−1n−1qn_Gh,1

n1,q/n1forn≥1.

Recently, Kim has studiedp-adic fermionic integral onZpconnected with the problems

of mathematical physicssee6,10,11, and our result are closely related to his results. In the future, we will try to study p-adic stochastic problems associated with our theorems. For example, p-adicq-Bernstein polynomials seem to be closely related to our resultssee

References

1 I. N. Cangul, H. Ozden, and Y. Simsek, “A new approach to q-Genocchi numbers and their interpolation functions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e793– e799, 2009.

2 J. Choi, P. J. Anderson, and H. M. Srivastava, “Someq-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of ordern, and the multiple Hurwitz zeta function,” Applied Mathematics

and Computation, vol. 199, no. 2, pp. 723–737, 2008.

3 L. Comtet, Advanced Combinatorics, D. Reidel, Dordrecht, The Netherlands, 1974.

4 E. Y. Deeba and D. M. Rodriguez, “Stirling’s series and Bernoulli numbers,” The American Mathematical

Monthly, vol. 98, no. 5, pp. 423–426, 1991.

5 M. Cenkci, M. Can, and V. Kurt, “p-adic interpolation functions and kummer-type congruences for

q-twisted euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 9, no. 2, pp. 203–216, 2004.

6 T. Kim, S. D. Kim, and D.-W. Park, “On uniform differentiability andq-Mahler expansions,” Advanced

Studies in Contemporary Mathematics, vol. 4, no. 1, pp. 35–41, 2001.

7 T. Kim, “Some identities on theq-Euler polynomials of higher order andq-Stirling numbers by the fermionicp-adic integral onZp,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009.

8 T. Kim, “The modified q-Euler numbers and polynomials,” Advanced Studies in Contemporary

Mathematics, vol. 16, no. 2, pp. 161–170, 2008.

9 T. Kim, “Note on the Eulerq-zeta functions,” Journal of Number Theory, vol. 129, no. 7, pp. 1798–1804, 2009.

10 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 2, pp. 288–299, 2002.

11 T. Kim, “Euler numbers and polynomials associated with zeta functions,” Abstract and Applied

Analysis, Article ID 581582, 11 pages, 2008.

12 T. Kim, “A note onp-adic q-integral onZpassociated withq-Euler numbers,” Advanced Studies in

Contemporary Mathematics, vol. 15, no. 2, pp. 133–137, 2007.

13 T. Kim, L.-C. Jang, and H. Yi, “A note on the modifiedq-bernstein polynomials,” Discrete Dynamics in

Nature and Society, vol. 2010, Article ID 706483, 12 pages, 2010.

14 T. Kim, J. Choi, and Y.-H. Kim, “Some identities on theq-Bernstein polynomials,q-Stirling numbers andq-Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 335–341, 2010.

15 T. Kim, “Power series and asymptotic series associated with theq-analog of the two-variablep-adic

L-function,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 186–196, 2005.

16 T. Kim, “Non-Archimedeanq-integrals associated with multiple Changheeq-Bernoulli polynomials,”

Russian Journal of Mathematical Physics, vol. 10, no. 1, pp. 91–98, 2003.

17 T. Kim, “q-Euler numbers and polynomials associated withp-adicq-integrals,” Journal of Nonlinear

Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007.

18 Q.-M. Luo, “q-extensions for the Apostol-Genocchi polynomials,” General Mathematics, vol. 17, no. 2, pp. 113–125, 2009.

19 Q.-M. Luo, “Fourier expansions and integral representations for Genocchi polynomials,” Journal of

Integer Sequences, vol. 12, no. 1, article 09.1.4, 2009.

20 Q.-M. Luo, “Some results for the q-Bernoulli and q-Euler polynomials,” Journal of Mathematical

Analysis and Applications, vol. 363, no. 1, pp. 7–18, 2010.

21 H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, “A note onp-adicq-Euler measure,” Advanced

Studies in Contemporary Mathematics, vol. 14, no. 2, pp. 233–239, 2007.

22 K.i Shiratani and S. Yamamoto, “On ap-adic interpolation function for the Euler numbers and its derivatives,” Memoirs of the Faculty of Science, Kyushu University A, vol. 39, no. 1, pp. 113–125, 1985.

23 Y. Simsek and M. Acikgoz, “A new generating function ofqBernstein-type polynomials and their interpolation function,” Abstract and Applied Analysis, Article ID 769095, 12 pages, 2010.

24 Y. Simsek, “Onp-Adic Twisted q-L-functions related to generalized twisted bernoulli numbers,”

25 Y. Simsek, “Theorems on twistedL-function and twisted Bernoulli numbers,” Advanced Studies in

Contemporary Mathematics, vol. 11, no. 2, pp. 205–218, 2005.

26 Y. Simsek, “q-Dedekind type sums related to q-zeta function and basic L-series,” Journal of

Mathematical Analysis and Applications, vol. 318, no. 1, pp. 333–351, 2006.

27 H. J. H. Tuenter, “A symmetry of power sum polynomials and Bernoulli numbers,” The American