Some Properties of Multiple Generalized q Genocchi Polynomials with Weight and Weak Weight

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Volume 2012, Article ID 179385,13pages doi:10.1155/2012/179385

Research Article

Some Properties of Multiple Generalized

q

-Genocchi Polynomials with Weight

α

and

Weak Weight

β

J. Y. Kang

Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Correspondence should be addressed to J. Y. Kang,[email protected]

Received 28 May 2012; Revised 21 August 2012; Accepted 22 August 2012

Academic Editor: Cheon Ryoo

Copyrightq2012 J. Y. Kang. 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.

The present paper deals with the variousq-Genocchi numbers and polynomials. We define a new type of multiple generalizedq-Genocchi numbers and polynomials with weightαand weak weight

βby applying the method ofp-adic q-integral. We will find a link between their numbers and polynomials with weightαand weak weightβ. Also we will obtain the interesting properties of their numbers and polynomials with weightαand weak weightβ. Moreover, we construct a Hurwitz-type zeta function which interpolates multiple generalizedq-Genocchi polynomials with weightαand weak weightβand find some combinatorial relations.

1. Introduction

Letpbe a fixed odd prime number. Throughout this paperZp,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 _see₁_–₂₁_{. When one talks of}_q_-extension,_q_{is variously considered as an}

indeterminate, a complexq ∈ C, or ap-adic number q ∈ Cp. If q ∈ C, then one normally assumes|q|<1. Ifq∈Cp, then we assume that|q−1|p<1.

Throughout this paper, we use the following notation:

x_q 1−q

x

1−q, x−q

1−−qx

1q . 1.1

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We say thatg:Zp → Cpis uniformly diﬀerentiable function at a pointa∈Zpand we writeg ∈UDZpif the diﬀerence quotientsΦg :Zp×Zp → Cpsuch that

Φg

x, y gx−g

y

x−y 1.2

have a limitgaasx, y → a, a.

Letdbe a fixed integer, and letpbe a fixed prime number. For any positive integerN, we set

XXdlim_← N

_Z

dpN_Z

, X1Zp,

X∗

0<a<dp

a,p1

adpZp,

adpNZpx∈X|x≡amoddpN ,

1.3

wherea∈Zlies in 0≤a < dpN_.

For any positive integerN,

μq

adpNZp q

a

dpN q

1.4

is known to be a distribution onX.

Forg∈UDZp, Kim defined theq-deformed fermionicp-adic integral onZp:

I−qg

Zpgxdμ−qx Nlim→ ∞

1

pN −q

pN₋₁

x0

gx−qx. 1.5

see1–13, and note that

Zpgxdμ−qx

X

gxdμ−qx. 1.6

We consider the case q ∈ −1,0 corresponding to q-deformed fermionic certain and annihilation operators and the literature given there in9,13,14.

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is odd. Then the multiple generalized Genocchi numbers,Gn,χr, and the multiple generalized

Genocchi polynomials,Gn,χrx, associated withχ, are defined by

Fχrt

⎛

⎝2taf−10χa−1

a_eat

eft₁

⎞ ⎠

r

∞

n0

Gn,χr

tn

n!,

Fχrt, x

⎛

⎝2tfa−10χa−1aeat

eft₁ e

tx

⎞ ⎠

r

∞

n0

Gn,χrx

tn

n!.

1.7

In the special case x 0,Gn,χr Gn,χr0are called thenth multiple generalized Genocchi

numbers attached toχ.

Now, having discussed the multiple generalized Genocchi numbers and polynomials, we were ready to multiple-generalize them to their q-analogues. In generalizing the generating functions of the Genocchi numbers and polynomials to their respective q -analogues; it is more useful than defining the generating function for the Genocchi numbers and polynomialssee12.

Our aim in this paper is to define multiple generalizedq-Genocchi numbersGn,χ,qα,β,rand

polynomials Gn,χ,qα,β,rx with weightαand weak weight β. We investigate some properties

which are related to multiple generalized q-Genocchi numbers Gn,χ,qα,β,r and polynomials

Gn,χ,qα,β,rx with weight α and weak weight β. We also derive the existence of a specific

interpolation function which interpolate multiple generalizedq-Genocchi numbers Gn,χ,qα,β,r

and polynomialsGn,χ,qα,β,rxwith weightαand weak weightβat negative integers.

2. The Generating Functions of Multiple Generalized

q

-Genocchi

Numbers and Polynomials with Weight

α

and Weak Weight

β

Many mathematicians constructed various kinds of generating functions of theq-Gnocchi numbers and polynomials by usingp-adicq-Vokenborn integral. First we introduce multiple generalizedq-Genocchi numbers and polynomials with weightαand weak weightβ.

Let us define the generalized q-Genocchi numbersGn,χ,qα,β and polynomials Gn,χ,qα,βx

with weightαand weak weightβ, respectively,

Fχ,qα,βt ∞

n0

Gn,χ,qα,β

tn

n!

X

tχxexqαt_dμ

−qβx,

Fχ,qα,βt, x ∞

n0

Gn,χ,qα,βx

tn

n!

X

tχyexyqαt_dμ

−qβ

y.

2.1

By using the Taylor expansion ofexqαt_{, we have}

∞

n0

X

χxxn_qαdμ_−qβxt

n

n!

∞

n0

Gn,χ,qα,β

tn−1

n! G

α,β

0,χ,q

∞

n0

G_nα,β₁_,χ,q

n1 tn

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By comparing the coeﬃcient of both sides oftn_/n_{! in}_2.2_{, we get}

G_nα,β₁_,χ,q

n1

2_qβ

1−qαn f−1

a0

−1aqβaχa

n

l0

n l

−1lqαal 1

1qfαlβ. 2.3

From2.2and2.3, we can easily obtain that

∞

n0

Gn,χ,qα,β

tn

n!

∞

n0

t

X

χxxn_qαdμ_−qβx

tn

n! 2qβt

∞

l0

−1lqβlχlelqαt_. _2.4

Therefore, we obtain

Fχ,qα,βt 2qβt

∞

l0

−1lqβlχlelqαt

∞

n0

Gn,χ,qα,β

tn

n!. 2.5

Similarly, we find the generating function of generalizedq-Genocchi polynomials with weightαand weak weightβ:

G₀α,β_,χ,qx 0, G

α,β

n1,χ,qx

n1

X

χyxyn_qαdμ−qβ

y 2_qβ

∞

l0

−1lqβlχlxln_qα.

2.6

From2.6, we have

Fχ,qα,βt, x 2qβt

∞

l0

−1lqβlχlexlqαt

∞

n0

Gn,χ,qα,βx

tn

n!. 2.7

Observe thatFχ,qα,βt Fχ,qα,βt,0. Hence we haveGn,χ,qα,β Gn,χ,qα,β0. Ifq → 1 into2.7, then

we easily obtainFχt, x.

First, we define the multiple generalizedq-Genocchi numbersGn,χ,qα,β,r with weightα

and weak weightβ:

Fχ,qα,β,rt 2r_qβtr

∞

k1,...,kr0

−1ri1ki_qβr_i1ki

_r

i1

χki

eri1kiqαt

tr

X · · ·

X

r-times

χx1· · ·χxrex1···xrqαtdμ−qβx₁· · ·dμ_−qβx_r

∞

n0

Gn,χ,qα,β,r

tn

n!.

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Then we have

∞

n0

X · · ·

X

r-times

χx1· · ·χxrx1· · ·xrnqαdμ_−qβx₁· · ·dμ_−qβx_r

tn

n!

∞

n0

Gn,χ,qα,β,r

tn−r n!

r−1

n0

Gn,χ,qα,β,r

tn−r n!

∞

n0

G_nα,β,r_r,χ,q

nr r r!

tn

n!,

2.9

wherenrr nr!/n!r!.

By comparing the coeﬃcients on the both sides of 2.9, we obtain the following theorem.

Theorem 2.1. Letq∈Cpwith|1−q|p<1andn∈Z. Then one has

G₀α,β,r_,χ,qG₁α,β,r_,χ,q· · ·G_r−α,β,r₁_,χ,q 0,

Gnα,β,rr,χ,q

nr r r!

X · · ·

X

r-times

χx1· · ·χxrx1· · ·xrnqαdμ_−qβx₁· · ·dμ_−qβx_r

2

r qβ

1−qαn f−1

a1,...,ar0

n

l0

n l

r

i1

χai

−1lri1ai_qαlβri1ai

1qfαlβr

2r_qβ

∞

m0

f−1

a1,...,ar0

mr−1 m

−1ri1aim_qβri1aifm×

_r

i1

χai

_r

i1

aifm

n

qα

.

2.10

From now on, we define the multiple generalizedq-Genocchi polynomialsGn,χ,qα,β,rx

with weightαand weak weightβ.

Fχ,qα,β,rt, x 2r_qβtr

∞

k1,...,kr0

−1ri1ki_qβr_i1ki

_r

i1

χki

eri1kixqαt

tr

X · · ·

X

r-times

χy1

· · ·χyr

exy1···yrqαt_dμ

−qβ

y1

· · ·dμ_−qβ

yr

∞

n0

Gn,χ,qα,β,rx

tn

n!.

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Then we have

∞

n0

X · · ·

X

r-times

χy1

· · ·χyr

xy1· · ·yr

n

qαdμ−qβ

y1

· · ·dμ_−qβ

yr

tn

n!

∞

n0

Gn,χ,qα,β,rx

tn−r n!

r−1

n0

Gn,χ,qα,β,rx

tn−r n!

∞

n0

G_nα,β,r_r,χ,qx

nr r r!

tn

n!,

2.12

wherenrr nr!/n!r!.

By comparing the coeﬃcients on the both sides of 2.12, we have the following theorem.

Theorem 2.2. Letq∈Cpwith|1−q|p<1andn∈Z. Then one has

G₀α,β,r_,χ,qx G₁α,β,r_,χ,qx · · ·G_r−α,β,r₁_,χ,qx 0,

Gnα,β,rr,χ,qx

nr r r!

X · · ·

X

r-times χy1

· · ·χyr

xy1· · ·yr

n

qαdμ−qβ

y1

· · ·dμ_−qβ

yr

2

r qβ

1−qαn f−1

a1,...,ar0

n

l0

n l

r

i1

χai

−1lri1ai_qαlxαlβri1ai

1qfαlβr

2r_qβ

∞

m0

f−1

a1,...,ar0

mr−1 m

−1ri1aim_qβri1aifm

×

_r

i1

χai

_r

i1

aifmx

n

qα

.

2.13

In2.11, we simply identify that

lim

q→1F

α,β,r

χ,q t, x 2rtr ∞

k1,...,kr0

−1ri1ki

_r

i1

χki

eri1kixt

⎛

⎝2tfa−10−1

a_χ_a_eat

1eft

⎞ ⎠

r

etx_Fr

χ t, x.

2.14

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3. Modified Multiple Generalized

q

-Genocchi Polynomials with

Weight

α

and Weak Weight

β

In this section, we will investigate about modified multiple generalizedq-Genocchi numbers and polynomials with weight α and weak weight β. Also, we will find their relations in multiple generalizedq-Genocchi numbers and polynomials with weightαand weak weight β.

Firstly, we modify generating functions of Gn,χ,qα,β,r and G α,β,r

n,χ,q x. We access some

relations connected to these numbers and polynomials with weightαand weak weight β. For this reason, we assign generating function of modified multiple generalizedq-Genocchi numbers and polynomials with weightαand weak weightβwhich are implied byGn,χ,qα,β,r

andGn,χ,qα,β,rx. We give relations between these numbers and polynomials with weightαand

weak weightβ.

We modify2.11as follows:

Fα,β,r

χ,q t, x Fχ,qα,β,r

q−αxt, x, 3.1

whereFχ,qα,β,rt, xis defined in2.11.

From the above we know that

Fα,β,r

χ,q t, x ∞

n0

q−nrαxGn,χ,qα,β,rx

tn

n!. 3.2

After some elementary calculations, we attain

Fα,β,r

χ,q t, x q−αrxeq

−αx_x

qαt_Fα,β,r

χ,q t, 3.3

whereFχ,qα,β,rtis defined in2.8.

From the above, we can assign the modified multiple generalized q-Genocchi polynomialsεn,χ,qα,β,rxwith weightαand weak weightβas follows:

Fα,β,r

χ,q t, x ∞

n0

εn,χ,qα,β,rx

tn

n!. 3.4

Then we have

εn,χ,qα,β,rx q−nrαxG α,β,r

n,χ,q x. 3.5

Theorem 3.1. Forr∈Nandn∈Z, one has

εn,χ,qα,β,rx q−nrαx n

i0

n i

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Corollary 3.2. Forr∈Nandn∈Z, by using3.7, one easily obtains

εn,χ,qα,β,rx q−nrαx ∞

m0

n

j0

n−j

l0

n j, l, n−j−l

n−jm−1 m

−1lqα{jlxm}G_j,χ,qα,β,r. 3.7

Secandly, by using generating function of the multiple generalized q-Genocchi polynomials with weight αand weak weight β, which is defined by2.11, we obtain the following identities.

By using2.13, we find that

Gnα,β,rr,χ,qx

nr r r!

2r_qβ

∞

m0

f−1

a1,...,ar0

mr−1 m

−1ri1aim

×qβri1aifm

_r

i1

χai

_r

i1

aifmx

n

qα

2r_qβ

f−1

a1,...,ar0

n

l0

l

a0

n a, l−a, n−l

−1ari1ai_q{αan−lβ}r_i1ai

×

_r

i1

χai

xn−l_qα

1−qαl₁_qf{αan−lβ}r.

3.8

Thus we have the following theorem.

Theorem 3.3. Letq∈Cpwith|1−q|p<1andr∈N. Then one has

Gnα,β,rr,χ,qx

nr r r!

2r_qβ

f−1

a1,...,ar0

n

l0

l

a0

n a, l−a, n−l

−1ari1ai_q{αan−lβ}ri1ai

×

_r

i1

χai

xn−l_qα

1−qαl₁_qf{αan−lβ}r.

3.9

By using2.13, we have

Fχ,qα,β,rt, x 2r_qβtr

∞

n0

n

l0

n l

₋₁l_qαlx

1−qn

f−1

a1,...,ar0

−1ri1ai

×qαlβri1ai

_r

i1

χai

_∞

m0

mr−1 m

−qfαlβ mt

n

n!.

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Thus we have

∞

n0

Gn,χ,qα,β,rx

tn

n!

∞

n0

2r_qβtr

n

l0

n l

−1lqαlx1−qα−n

f−1

a1,...,ar0

−1ri1ai

×qαlβri1ai

_r

i1

χai

1qfαlβ −rtn

n!.

3.11

By comparing the coeﬃcients of both sides of nr!/tnr _{in the above, we arrive at the}

following theorem.

Theorem 3.4. Letq∈Cpwith|1−q|p<1,r∈N. Then one has

Gnα,β,rr,χ,qx

nr r r!

2r_qβ

n

l0

n l

−1lqαlx1−qα−n

f−1

a1,...,ar0

−1ri1ai

×qαlβri1ai

_r

i1

χai

1qfαlβ −r.

3.12

From2.12, we easily know that

∞

n0

Gn,χ,qα,β,rx

tn

n!

∞

n0

2r_qβ

nr r

r!

∞

k1,...,kr0

−1ri1ki_qβr_i1ki

_r

i1

χki

×

x

r

i1

ki

n

qα

tnr

nr!.

3.13

From the above, we get the following theorem.

Theorem 3.5. Letr∈N,k∈Z. Then one has

G₀α,β,r_,χ,qx G₁α,β,r_,χ,qx · · ·G_r−α,β,r₁_,χ,qx 0,

G_lα,β,r_r,χ,qx 2r_qβ

lr r

r!

∞

k1,...,kr0

−1ri1ki_qβri1ki

_r

i1

χki

x

r

i1

ki

l

qα

.

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From2.13, we have

∞

n0

Gn,χ,qα,β,rx

tn

n!

∞

n0

Gn,χ,qα,β,sx

tn

n!

2r_qβstrs

f−1

a1,...,ar0

∞

m0

mr−1 m

−1ri1aim_qβri1aifm

×

_r

i1

χai

eri1aifmxqαt

f−1

b1,...,bs0

∞

k0

ks−1 k

−1si1bik

×qβsi1bifk

_s

i1

χbi

eri1bifkxqαt_.

3.15

By using Cauchy product in3.15, we obtain

∞

n0

n

j0

n j

G_j,χ,qα,β,rxG_n−j,χ,qα,β,sxt

n

n!

2r_qβstrs

∞

n0

n

j0

f−1

a1,...,ar0

f−1

b1,...,bs0

jr−1 j

n−js−1 n−j

×−1ri1aisi1bin_qβri1aisi1bifn

_r

i1

χai

_s

i1

χbi

×eri1aifjxqαt_e s

i1bifn−jxqαt_.

3.16

From3.16, we have

∞

m0 ⎛

⎝m

j0

m j

G_j,χ,qα,β,rxG_m−j,χ,qα,β,s x

⎞

⎠tm

m!

∞

m0

2r_qβstrs

∞

n0

n

j0

f−1

a1,...,ar0

f−1

b1,...,bs0

jr−1 j

n−js−1 n−j

_r

i1

χai

_s

i1

χbi

×

⎛ ⎝

_r

i1

aifjx

qα

_s

i1

bifn−j x

qα ⎞ ⎠ m tm

m!.

3.17

By comparing the coeﬃcients of both sides of tmrs_/_m_r _s_{! in} _3.17_{, we have the}

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Theorem 3.6. Letr∈Nands∈Z. Then one has

lrs

j0

lrs

j G

α,β,r j,χ,q xG

α,β,s lrs−j,χ,qx

_l_r_s

l

rs!

2r_qβs

∞

n0

n

j0

f−1

a1,...,ar0

f−1

b1,...,bs0

jr−1 j

n−js−1 n−j

_r

i1

χai

_s

i1

χbi

× ⎛ ⎝ _r

i1

aifjx

qα

_s

i1

bif

n−jx

qα ⎞ ⎠ l . 3.18

Corollary 3.7. In3.18settings1, one has

lr1

j0

lr1

j G

α,β,r j,χ,q xG

α,β,1

lr1−j,χ,qx

_l_r₁

l

r1!

2r_qβ1

∞

n0

n

j0

f−1

a1,...,ar0

f−1

b10

jr−1 j

−1ri1aib1n_qβri1aib1fn

×

χb1

r

i1

χai

⎛ ⎝

_r

i1

aifjx

qα

b1f

n−jx_qα ⎞ ⎠

l

.

3.19

By using2.13we have the following theorem.

Theorem 3.8. Distribution theorem is as follows:

Gnα,β,rr,χ,q

fn_qα

fr_−qβ

f−1

a1,...,ar0

−1ri1ai_qβr_i1ai

_r

i1

χai

G_nα,β,r_r,qf

a1· · ·ar

f

,

Gnα,β,rr,χ,qx

fn_qα

fr_−qβ

f−1

a1,...,ar0

−1ri1ai_qβr_i1ai

_r

i1

χai

G_nα,β,r_r,qf

xa1· · ·ar

f

.

3.20

4. Interpolation Function of Multiple Generalized

q

-Genocchi

Polynomials with Weight

α

and Weak Weight

β

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Let us define interpolation function of theG_kα,β,r_r,qxas follows.

Definition 4.1. Letq, s∈Cwith|q|<1 and 0< x≤1. Then one defines

ζχ,qα,β,rs, x 2rqβ

∞

k1,...,kr0

−1ri1ki_qβr

i1kir

i1χki

xr_i₁ki

s

qα

. 4.1

We callζqα,β,rs, xthe multiple generalized Hurwitz typeq-zeta funtion.

In4.1, settingr1, we have

ζχ,qα,β,1s, x 2qβ

∞

l0

−1lqβlχl

xls_qα

ζχ,qα,βs, x. 4.2

Remark 4.2. It holds that

lim

q→1ζ

α,β,r

χ,q s, x 2r ∞

k1,...,kr0

−1ri1kir

i1χki

xr_i₁ki

s . 4.3

Substitutings−n, n∈Zinto4.1, then we have,

ζχ,qα,β,r−n, x 2r_qβ

∞

k1,...,kr0

−1ri1ki_qβri1ki

_r

i1

χki

x

r

i1

ki

n

qα

. 4.4

Setting3.14into the above, we easily get the following theorem.

Theorem 4.3. Letr∈N,n∈Z. Then one has

ζχ,qα,β,r−n, x

G_nα,β,r_r,χ,qx

nr r r!

. 4.5

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