A Note on h, q -Genocchi Polynomials and Numbers of Higher Order

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doi:10.1155/2010/309480

Research Article

A Note on h, q -Genocchi Polynomials and Numbers of Higher Order

Lee-Chae Jang,

¹

Kyung-Won Hwang,

²

and Young-Hee Kim

³

1Department of Mathematics and Computer Science, KonKuk University, Chungju 138-701, South Korea

2Department of Mathematics, Dong-A University, Busan 604-714, South Korea

3Division of General Education, Kookmin University, Seoul 136-702, South Korea

Correspondence should be addressed to Lee-Chae Jang,[email protected] Received 10 July 2009; Revised 5 December 2009; Accepted 11 February 2010 Academic Editor: Patricia J. Y. Wong

Copyrightq2010 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 investigate several arithmetic properties ofh, q-Genocchi polynomials and numbers of higher order.

1. Introduction and Preliminaries

Recently, Kim 1 studied q-Genocchi and Euler numbers using Fermionic q-integral and introduced related applications. Kim 2 also gives theq-extensions of the Euler numbers which can be viewed as interpolating of q-analogue of Euler zeta function at negative integers and gives Bernoulli numbers at negative integers by interpolating Riemann zeta function. These numbers are very useful for number theory and mathematical physics.

Kim 3, 4 studied q-Bernoulli numbers and polynomials related to Gaussian binomial coeﬃcient and studied also some identities ofq-Euler polynomials andq-stirling numbers.

Kim5made Dedekind DC sum in the meaning of extension of Dedekind sum or Hardy sum and introduced lots of interesting results. The purpose of this paper is to investigate several arithmetic properties of h, q-Genocchi polynomials and numbers of higher order.

Letpbe a fixed odd prime. Throughout this paperZ,Zp,Qp, andCpwill, respectively, denote the ring of rational integers, the ring ofp-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure ofQp. Letvpbe the normalized exponential valuation ofCp with |p|p p^−v^p^p p⁻¹.When one talks ofq-extension, qis variously considered as an indeterminate, a complex number q ∈ C,or a p-adic number q∈Cp. Ifq∈C,one normally assumes|q|<1.Ifq∈Cp,then we assume|q−1|_p< p^−1/p−1,

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so thatq^xexpxlogqfor|x|_p≤1. We also use the notations

x_q 1−q^x

1−q, x_−q 1−

−q_x

1 q 1.1

for allx∈Zpsee5–12. Hence, limq→1x_qx.

Letdbe a fixed positive integer withp, d 1. We now set Xlim_←−

N

Z

dp^NZ, X^∗

0<a<dp a,p1

a dpZp, a dp^NZp

x∈X|x≡a

modp^N , 1.2

wherea∈Zlies in 0≤a < dp^N. For anyN∈N, we set

µq

a dp^NZp

−q_a dp^N

−q

, 1.3

and this can be extended to a distribution onZp.

We say thatf is a uniformly diﬀerentiable function at a pointa∈ Zp and writef ∈ UDZp, if the diﬀerence quotientsFfx, y fx−fy/x−yhave a limitfaas x, y → a, a cf.13–23.

Forf∈UDZp, thep-adic invariant integral onZpis defined as

I f

X

fxdµx lim

N→ ∞ dp^N−1

x0

fx−1^x 1.4

see14,23. Letn∈Nandfnx fx n. From1.4, we have

I fn

I f

2 n−1

l0

−1^lfl. 1.5

Thep-adic integral has been used in many areas such as mathematics, physics, probability theory, dynamical systems, and biological models. Especially, Khrennikov24–26applied to many areas using ingenious technique. The Genocchi numbersGn and polynomialsGnx are defined by the generating functions as follows:

2t e^t 1 ^∞

n0

Gntⁿ

n!, 2t

e^t 1e^xt^∞

n0

Gnxtⁿ

n! 1.6

see5,7,15. Theq-extension of Genocchi numbers are defined by Fqt t

∞ m0

−1^mq^me^m^q^t^∞

n0

Gn,qtⁿ

n! 1.7

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see1,2, and theq-extension of Genocchi polynomials is also given by

Gn,qx ⁿ

l0

n l

q^lxGl,qx^n−l_q . 1.8

InSection 2, we investigate several arithmetic properties ofh, q-Genocchi polynomials and numbers of higher order.

2. h, q-Genocchi Numbers of Higher Order

Leth, k ∈Nandq∈Cwith|q−1|_p< p^−1/p−1. Theh, q-Genocchi polynomialsG^h,km,q xof orderkare defined as

t^k

Zp

· · ·

Zp

e^x¹^···x^k^x^q^tq^h−1x¹^{··· h−kx}^kdµqx1· · ·dµqxk

∞

m0G^h,km,q xt^m

m!, 2.1

where

Zpfxdµqx limN→ ∞1/p^N_−qp^N−1

x0 fx−q^x. It is easily to see thatG^h,k_0,q x

· · ·G^h,k_k−1,qx 0 for eachh∈Zandk∈N. From2.1, we can obtain the following theorem.

Theorem 2.1. Leth∈Zandm, k∈N. Then for allx∈Zp,

k!

m k!G^h,k_{m k,q}x

Zp

· · ·

Zp

x1 · · ·xk x^m_qq^h−1x¹^{··· h−kx}^kdµqx1· · ·dµqxk. 2.2

FromTheorem 2.1, if we takek−mm >0, then

1 m!_{m k}

m

G^−m,k_{m k,q}x

Zp

· · ·

Zp

x1 · · ·xk x^m_qq^{−m 1x}¹^{−···−m kx}^kdµqx1· · ·dµqxk. 2.3

Now, we defineh, q-Genocchi number of higher order as follows:

G^−m,km,q G^−m,km,q 0. 2.4

From2.4, we can derive the following theorem.

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Theorem 2.2. Leth∈Zandm, k∈N. Then one has

G^−m,k_{m k,q} m!_{m k}

m

lim

N→ ∞

1 p^N_k

−q p^N−1

x10

· · ·

p^N−1 xk0

−1^x¹^{··· x}^kx1 · · · xk^m_qq^−x¹^{m−···−x}^k^{m k−1}

2^k_q 1−q_m^m

i0

m i

−1ⁱ 1 1 q^i−m

· · ·

1 q^{i−m−k 1},

2.5

where_m

i

m· · ·m−k 1/k!.

Note that lim_q_→₁G^−m,km,q G^km , whereG^km are the ordinary Genocchi numbers of order kdefined as

2t e^t 1

_k ^∞

n0

G^kn tⁿ

n!. 2.6

By2.4and2.5, we can obtain the following theorem.

Theorem 2.3. Letm∈N. Then one has

G^−m,1_{m 1,q} m 1 ^m

i0

m i

q^xⁱG^−m,1_{i 1,q}

i 1 x^m−i_q 2_q 1−qm

m j0

q^ix m

i

−1^j

1 q^j−m. 2.7

It is easily to check that G^−n,1_{n 1,q}

n 1

Zp

q^{−n 1t}x tⁿ_qdµqt

1 q

1 q^ddⁿ_q^d−1

x0

−1ⁱq⁻ⁿⁱ

Zp

q^{−n 1dt} x i

d t

n q^d

dµq^dt,

2.8

wheren, d∈Nwithd≡1mod2. Thus we have the following theorem.

Theorem 2.4. Letd, n∈Nwithd≡1 mod 2. Then for allx∈Zp,

G^−n,1_{n 1,q}x

n 1 1 q

1 q^ddⁿ_qⁿ

i0

−1ⁱq⁻ⁿⁱG^−n,1_{n 1,q} x i

d

. 2.9

We note that if we takex0, then we have G^−m,1_{m 1,q}

m 1 1 q 1 q^m

m k0

m k

n^k_qG^−m,1_{k 1,q}n

k 1

n−1

j0

−1^jq^−m−kj j_m−k

q , 2.10

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wheren1mod 2. By2.10, we easily see that G^−m,1_{m 1,q}

m 1 − 1 q

1 qⁿn^m_q G^−m,1_{m 1,q}

m 1 1 q 1 q^m

m−1

k0

n^k_qG^−m,1_{k 1,q}n

k 1 n−1

j0

−1^jq^−m−kj j_m−k

q . 2.11

Note that limq→1G^−m,1m,q Gm, whereGmare themth Genocchi numbers defined as 2t

e^t 1 ^∞

n0

Gntⁿ

n!. 2.12

From2.11, we can see that

1−n^mGm 1

m 1 ^m−1

k0

m k

n^kGk 1

k 1

n−1

j0

−1^jj^m−k. 2.13

LetFqt, xbe the generating function ofG^−m,1m,q as follows:

Fqt, x ^∞

n0

G^−n,1_{n 1,q}x n 1

tⁿ

n!. 2.14

By2.7and2.14, we see that

Fqt, x ^∞

k0

1 q^∞

n0

−1ⁿq^−knn x^k_q t^k

k!

1 q^∞

n0

−1ⁿ^∞

n0

−1ⁿ^∞

k0

q^−knn x^k_qt^k k!

1 q^∞

n0

−1ⁿe^{n x}^q^q⁻ⁿ^t.

2.15

By2.14and2.15, we can obtain the following theorem.

Theorem 2.5. Letm∈N. Then for allx∈Zp,

G^−m,1_{m 1,q}x

m 1 2_q^∞

n0

q^−nm−1ⁿn x^m_q. 2.16

Acknowledgment

This paper was supported by KOSEF2009-0073396, 2009-A419-0065.

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