On Multiple Interpolation Functions of the Genocchi Polynomials

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Volume 2010, Article ID 351419,13pages doi:10.1155/2010/351419

Research Article

On Multiple Interpolation Functions of

the

q

-Genocchi Polynomials

Seog-Hoon Rim,

1

_{Jeong-Hee Jin,}

2

_{Eun-Jung Moon,}

2

and Sun-Jung Lee

2

1_{Department of Mathematics Education, Kyungpook National University, Tagegu 702-701, South Korea} 2_{Department of Mathematics, Kyungpook National University, Tagegu 702-701, South Korea}

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

Received 14 December 2009; Accepted 29 March 2010

Academic Editor: Wing-Sum Cheung

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.

Recently, many mathematicians have studied various kinds of the q-analogue of Genocchi numbers and polynomials. In the workNew approach toq-Euler, Genocchi numbers and their interpolation functions, “Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 105– 112, 2009.”, Kim defined new generating functions ofq-Genocchi,q-Euler polynomials, and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz typeq -zeta function. This function interpolatesq-Genocchi polynomials at negative integers. Finally, we also give some identities related to these polynomials.

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−vpp₁_/p_see₁_.

When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ Cor ap-adic number q ∈ Cp. If q ∈ C, then one normally assumes|q| < 1. Ifq∈Cp, then we assume that|q−1|_p<1. In this paper, we use the following notation:

x x:q 1−q

x

1−q, x−q

1−−qx

1q 1.1

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We say thatf :Zp → Cpis uniformly diﬀerentiable function at a pointa∈Zpand we writef ∈UDZpif the diﬀerence quotientsΦf :Zp×Zp → Cpsuch that

Φf

x, y fx−f

y

x−y 1.2

have a limit fa as x, y → a, a. For f ∈ UDZp, the q-deformed fermionic p-adic integral is defined as

I−q

f

Zp

fxdμ−qx lim N→ ∞

1

pN

−q pN₋₁

x0

fx−qx 1.3

see4–6. Note that

I−1

flim q→1I−q

f

Zp

fxdμ−1x 1.4

see7–9. Letf1xbe the translation withf1x fx1.Then we have the following

integral equation:

I−1

f1

I−1

f2f0, 1.5

see10–12.

The ordinary Genocchi numbers and polynomials are defined by the generating functions as, respectively,

Ft 2t et₁

∞

n0

Gnt n

n!, |t|< π,

Ft, x 2t et₁e

xt∞ n0

Gnxt n

n!, |t|< π.

1.6

Observe thatGn0 Gnsee10,11,13.

These numbers and polynomials are interpolated by the Genocchi zeta function and Hurwitz-type Genocchi zeta function, respectively,

ζGs 2

∞

n1

−1n

ns , s∈C,

ζGs, x 2

∞

n0

−1n

nxs, s∈C, 0< x≤1.

1.7

Thus we note that Genocchi zeta functions are entire functions in the whole complexs-plane

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Various kinds of theq-analogue of the Genocchi numbers and polynomials, recently, have been studied by many mathematicians. In this paper, we use Kim’s14–16methods. By using p-adic q-Vokenborn integral 6, Kim 2, 7–9, 14–18 constructed many kind of generating functions of the q-Euler numbers and polynomials and their interpolation functions. He also gave many applications of these numbers and functions. He14definedq -extension Genocchi polynomials of higher order. He gave many applications and interesting identities. We give some of them in what follows.

Let q ∈ Cwith |q| < 1. Theq-Genocchi numbersGn,q and polynomials Gn,qxare defined by Kim of the generating functions as, respectively,

t

Zp

extdμ−qx

∞

n0

Gn,qt n

n!, |t|< π

t

Zp

exytdμ−q

y ∞

n0

Gn,qxt n

n!, |t|< π

1.8

see1,8–11,13,14,17. By using the Taylor expansion ofext,

∞

n0

Zp

xndμ−qxt n

n!

∞

n0

Gn,qt n−1

n! G0,q

∞

n0

Gn1,q

n1

tn

n!. 1.9

By comparing the coeﬃcient of both sides oftn_/n_{! in the above,}

G0,q0,

Gn1,q

n1

Zp

xndμ−qx

2

1−qn

n

l0

n l −1

l 1 1ql1.

1.10

From the above, we can easily derive that

∞

n0

Gn,qt n

n!

∞

n0

t

Zp

xndμ−qx t n

n!

∞

n0

t 2

1−qn

n

l0

n l −1

l 1 1ql1

tn

n!

2t ∞

m0

−1mqm_emt_.

1.11

Thus we have, following that,

Fqt 2t

∞

m0

−1mqmemt ∞

n0

Gn,qt n

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Using similar method to the above, we can find that

G0,qx 0,

Gn1,qx

n1

Zp

xyndμ−q

y 2

1−qn

n

l0

n l −1

l

qlx 1

1ql1.

1.13

Thus we can easily derive that

Fqt, x 2t

∞

m0

−1mqmemxt ∞

n0

Gn,qxt n

n!. 1.14

Observe thatFqt Fqt,0. Hence we haveGn,q0 Gn,q. Ifq → 1 into1.14, then we easily obtainFt, xin1.6.

Letq∈Cwith|q|<1,r ∈N, andn≥0. We now define as the generating functions of higher orderq-extension Genocchi numbersGn,qrand polynomialsGn,qrx, respectively,

Fqrt tr

Zp

· · ·

Zp

rtimes

ex1···xrt_dμ−

qx1· · ·dμ−qxr

∞

n0

Gn,qrt n

n!,

Fqrt, x tr

Zp

· · ·

Zp

rtimes

exx1···xrt_dμ−

qx1· · ·dμ−qxr

∞

n0

Gn,qrxt n

n!.

1.15

Then we have

∞

n0

Zp

· · ·

Zp

x1· · ·xrndμ−qx1· · ·dμ−qxr t n

n!

∞

n0

Gn,qrt n−r

n! r−1

n0

Gn,qrt n−r

n!

∞

n0

Gnrr,q _n_r

r

r!

tn

n!,

1.16

wheren_rr nr!/n!r!.

By comparing the coeﬃcient of both sides oftn_/n_{! in the above, we can derive that}

G₀r_,qG₁r_,q· · ·G_rr₋₁_,q0, Gnrr,q

_n_r r

r!

Zp

· · ·

Zp

x1· · ·xrndμ−qx1· · ·dμ−qxr

2r

1−qn

n

l0

n l −1

l 1 1ql1r.

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Therefore we obtain

Fqrt 2rtr

∞

m0

−1mqm

mr−1

m e

mt∞

n0

Gn,qrt n

n!. 1.18

Using similar method to the above, we can also derive that

G₀r_,qx G₁r_,qx · · ·G_rr₋₁_,qx 0, Gn,qrx

_n_r r

r!

2r

1−qn

n

l0

n l −1

l_qlx 1 1ql1r.

1.19

Thus we can easily obtain the following theorem.

Theorem 1.1. Forr∈Nandn≥0, one has

Fqrt, x 2rtr

∞

m0

−1mqm

mr−1

m e

mxt∞

n0

Gn,qrxt n

n!. 1.20

It is noted that ifr 1, then1.20reduces to1.14.

Remark 1.2. In1.20, we easily see that

lim q→1F

r

q t, x 2rtr

∞

m0

−1m

mr−1

m e

mxt

2rtretx ∞

m0

mr−1

m

−etm

2rtretx 1etr

Frt, x.

1.21

From the above, we obtain generating function of the Genocchi numbers of higher order. That is

Frt, x 2

r_tr_etx

1etr

∞

n0

Gnrxt n

n!. 1.22

Thus we have

lim q→1G

r

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Hence we have

Frt, x

2t et₁

· · ·

2t et₁

rtimes

etx

2rtretx ∞

n10

−1n1_en1t

∞

n20

−1n2_en2t· · ·

∞

nr0

−1nr_enrt

2rtretx ∞

n1,n2,...,nr0

−1n1n2···nr_en1n2···nrt

∞

n0

Gnrxt n

n!.

1.24

In14, Kim defined new generating functions ofq-Genocchi,q-Euler polynomials and their interpolation functions. In this paper, we give another definition of the multiple Hurwitz typeq-zeta function. This function interpolatesq-Genocchi polynomials at negative integers. Finally, we also give some identities related to these polynomials.

2. Modified Generating Functions of Higher Order

q

-Genocchi

Polynomials and Numbers

In this section, we study modified generating functions of the higher order q-Genocchi numbers and polynomials. We obtain some relations related to these numbers and polynomials. Therefore we define generating function of modified higher orderq-Genocchi polynomials and numbers, which are denoted byGn,qrxandGn,qr, respectively, in1.15. We give relations between these numbers and polynomials.

We modify1.20as follows:

Fr

q t, x Fqr

q−xt, x, 2.1

whereFqrt, xis defined in1.20. From the above we find that

Fr q t, x

∞

n0

q−nrxGn,qrxt n

n!. 2.2

After some elementary calculations, we obtain

Fr

q t, x q−rxexp

xq−xtFqrt, 2.3

where

Fqrt 2rtr

∞

m0

−1mqm

rm−1

m e

mt∞

n0

Gn,qrt n

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From the above, we can define the modified higher orderq-Genocchi polynomialsεn,qrxas follows

Fr

q t, x

∞

n0

εn,qrxt n

n! 2.5

Then we have

εn,qrx q−nrxGn,qrx. 2.6

By using Cauchy product in2.3, we arrive at following theorem.

εn,qrx q−nrx n

j0

n j q

jx_xn−j_Gr

j,q. 2.7

By using2.7, we easily obtain the following result.

Corollary 2.2. Forr∈N, andn≥0, one has

εn,qrx q−nrx

∞

m0

n

j0

n−j

l0

n j, l, n−j−l

n−jm−1

m −1

l

qmxjlG_j,qr. 2.8

We now give some identity related to the Genocchi polynomials and numbers of higher order.

Substitutingx0 into1.24, we find that

Gnr2rtr

∞

n1,n2,...,nr0

∞

j1,j2,...,jr0

j1j2···jrn

n j1, j2, . . . , jr

−1n1n2···nr

r

k0

njk

k. 2.9

By1.24and2.8, we arrive at the following theorem.

Gnr n

j0

n j −x

n−j

G_jrx. 2.10

By using1.24, we easily arrive at the following result.

Corollary 2.4. Forr, v∈Nandn≥0, one has

Grx Gvyn

n

j0

n j x

n−j_Grv

j

y, 2.11

whereGr_xn_{is replace by}_Gr

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3. Interpolation Function of Higher Order

q

-Genocchi Polynomials

Recently, higher order Bernoulli polynomials, Euler polynomials, and Genocchi polynomials have been studied by many mathematicians. Especially, in this paper, we study higher order Genocchi polynomials which constructed by Kim 15and see also the references cited in each of the these earlier works.

In14, by using the fermionicp-adic invariant integral onZp, the set ofp-adic integers, Kim gave a new construction of q-Genocchi numbers, Euler numbers of higher order. By usingq-Genocchi, Euler numbers of higher order, he investigated the interesting relationship betweenw-q-Euler polynomials andw-q-Genocchi polynomials. He also defined the multiple

w-q-zeta functions which interpolateq-Genocchi, Euler numbers of higher order.

By using similar method to that in the papers given by Kim14, in this section, we give interpolation function of the generating functions of higher orderq-Genocchi polynomials. From1.20, we easily see that

∞

k0

G_k,qrxt

k

k!

∞

k0

2rr!

kr r

∞

m0

−1mqm

mr−1

m mx

k tkr

kr!. 3.1

From the above we have

G_kr_r,qx 2rr!

kr r

∞

m0

−1mqm

mr−1

m mx

k_,

G₀r_,qx G₁r_,qx · · ·G_rr₋₁_,qx 0.

3.2

Hence we have obtain the following theorem.

Theorem 3.1. Letr, k∈Z. Then one has

G_kr_r,qx 2rr!

kr r

∞

m0

−1mqm

mr−1

m mx

k

. 3.3

Let us define interpolation function of theG_kr_r,qxas follows.

Definition 3.2. Letq, s∈Cwith|q|<1 and 0< x≤1. Then we define

ζqrs, x 2r

∞

n0

nr−1

n

−1nqn

nxs. 3.4

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Remark 3.3. It holds that

lim q→1ζ

r

q s, x 2r

∞

n0

nr−1

n

−1n

nxs. 3.5

From1.24, we easily see that

ζrs, x 2r

∞

n1,n2,...,nr0

−1n1n2···nr r

j1njx

s, 3.6

wheres∈C.

The functions in 3.5 and 3.6 interpolate the same numbers at negative integers. That is, these functions interpolate higher orderq-Genocchi numbers at negative integers. So, by3.5, we modify3.6in sense ofq-analogue.

In3.5and3.6, settingr1, we have

ζ1s, x 2

∞

n0

−1n

nxs ζGs, x, 3.7

where ζGs, x denotes Hurwitz type Genocchi zeta function, which interpolates classical Genocchi polynomials at negative integers.

Substitutings−k,k∈Zinto3.4. Then we have

ζqr−k, x 2r

∞

n0

rn−1

n −1

n

qnnxk. 3.8

Setting3.3into the above, we easily arrive at the following result.

Theorem 3.4. Letr, k∈Z. Then one has

ζqr−n, x

Gnrr,qx

r!n_rr. 3.9

4. Some Relations Related to Higher Order

q

-Genocchi Polynomials

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By using1.20, we find that

G_kr_r,qx

r!kr r

2r

∞

m0

−1mqm

mr−1

m

m qmxk

2r

∞

m0

−1mqm

mr−1

m

k

j0

k j m

j_qmk−j_xk−j

2r

∞

m0

−1mqm

mr−1

m

k

j0

k j

1−qmj

1−qj q

mk−j_xk−j

2r

∞

m0

−1mqm

mr−1

m

k

j0

k j

j

a0

j a

−1aqmak−j_xk−j

1−qj

2r k

j0

j

a0

k j

j a

−1axk−j

1−qj ∞

m0

−1mqm

mr−1

m q

mak−j

2r k

j0

j

a0

k j

j a

−1axk−j

1−qj1qak−j1r

2r k

j0

j

a0

−1a

k a, j−a, k−j

xk−j

1−qj1qak−j1r.

4.1

Thus we have the following theorem.

Theorem 4.1. Letq∈Cwith|q|<1. Letrbe a positive integer. Then one has

G_kr_r,qx

r!kr r

2r k

j0

j

a0

−1a

k a, j−a, k−j

xk−j

1−qj1qak−j1r. 4.2

By using1.20, we have

Fqrt, x 2rtr

∞

m0

−1mqm

mr−1

m e

mxt

2rtr ∞

m0

∞

n0

−1mqm

mr−1

m

1−qmx 1−q

n

tn

n!

2rtr ∞

m0

∞

n0

−1mqm

mr−1

m

1

1−qn

n

j0

n j

−qmxjt

n

n!

2rtr ∞

n0

n

j0

n j

−1jqjx

1−qn ∞

m0

mr−1

m −1

m_qj1mtn

n!.

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Thus we have

∞

n0

Gn,qrxt n

n!

∞

n0

2rtr

n

j0

n j −1

j

qjx1qj1−r1−q−nt

n

n!. 4.4

By comparing the coeﬃcients tn_/n_{! of both sides in the above, we arrive at the following} theorem.

Theorem 4.2. Letq∈Cwith|q|<1. Letrbe a positive integer. Then one has

Gnrr,qx

r!n_rr 2r

n

j0

n j −1

j

qjx1qj1−r1−q−n. 4.5

By using1.20, we have

∞

n0

Gn,qrxt n

n!

∞

n0

Gn,qyxt n

n!

2rytry ∞

n0

−1nqn

nr−1

n e

nxt∞ n0

−1nqn

ny−1

n e

nxt_.

4.6

By using Cauchy product into the above, we obtain

∞

n0

n

j0

n j G

r

j,qxG

r

n−j,q

yt

n

n!

2rytry ∞

n0

n

j0

n j −1

j

qj

jr−1

j −1

n−j

qn−j

n−jy−1

n−j e

jxt_en−jxt_.

4.7

From the above, we have

∞

m0

⎛ ⎝m

j0

m j G

r

j,qxG

r

m−j,q

y

⎞ ⎠tm

m!

∞

m0

⎛ ⎝₂ry

try ∞

n0

n

j0

−1nqn

jr−1

j

n−jy−1

n−j

jxn−jxm

⎞ ⎠tm

m!.

4.8

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Theorem 4.3. Letr, y∈Z. Then one has

kry j0

kry

j

G_j,qrxG_ky_r_y₋_j,qx

ry!kry k

2ry

∞

n0

n

j0

−1nqn

jr−1

j

n−jy−1

n−j

jxn−jxk.

4.9

Remark 4.4. In4.9settingy1, we have

kr1

j0

_k_r₁ j

G_j,qrxG_k1_r₁₋_j,qx

r1!kr1 k

2r1

∞

n0

n

j0

−1nqn

jr−1

j

jxn−jxk.

4.10

References

1 T. Kim, L. C. Jang, S. H. Rim, et al.,Introduction to Non-Archimedean Analysis, Kyo Woo Sa, Seoul, Korea, 2004,http://www.kyowoo.co.kr.

2 T. Kim, “A note onp-adicq-integral onZp associated withq-Euler numbers,”Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 133–138, 2007.

3 T. Kim, “A note on theq-Genocchi numbers and polynomials,”Journal of Inequalities and Applications, vol. 2007, Article ID 71452, 8 pages, 2007.

4 T. Kim, “q-Euler numbers and polynomials associated withp-adicq-integrals,”Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007.

5 T. Kim, “q-generalized Euler numbers and polynomials,”Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 293–298, 2006.

6 T. Kim, “q-Volkenborn integration,”Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.

7 T. Kim, “On aq-analogue of thep-adic log gamma functions and related integrals,”Journal of Number Theory, vol. 76, no. 2, pp. 320–329, 1999.

8 T. Kim, “Note on Dedekind type DC sums,”Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 249–260, 2009.

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

10 S.-H. Rim and T. Kim, “A note onp-adic Euler measure onZp,”Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 358–361, 2006.

11 S.-H. Rim and T. Kim, “A note onq-Euler numbers associated with the basicq-zeta function,”Applied Mathematics Letters, vol. 20, no. 4, pp. 366–369, 2007.

12 T. Kim, “The modified q-Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 161–170, 2008.

13 S.-H. Rim, K. H. Park, and E. J. Moon, “On Genocchi numbers and polynomials,”Abstract and Applied Analysis, vol. 2008, Article ID 898471, 7 pages, 2008.

14 T. Kim, “New approach toq-Euler, Genocchi numbers and their interpolation functions,”Advanced Studies in Contemporary Mathematics, vol. 18, no. 2, pp. 105–112, 2009.

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16 T. Kim, “On theq-extension of Euler and Genocchi numbers,”Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1458–1465, 2007.

17 T. Kim, “Euler numbers and polynomials associated with zeta functions,” Abstract and Applied Analysis, vol. 2008, Article ID 581582, 11 pages, 2008.