2. q -Extensions of Genocchi Numbers and Polynomials of N ¨orlund Type

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^−vpp1/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:

x_q 1−q^x

1−q, x_−q 1−

−qx

1q , 1.1

see1–10. Hence limq→1x_qxfor allx∈Zp.

Theq-factorial is defined asn_q! n_qn−1_q· · ·2_q1_q, and the Gaussian binomial coefficient is defined by the standard rule

n k

q

n_q!

n−k_q!k_q! n_qn−1_q· · ·n−k1_q!

k_q! , 1.2

see7,9. Note that limq→1ⁿ_kq ⁿ_k n!/n−k!k!nn−1· · ·n−k1/k!. It readily follows from1.2that

n1 k

q

n

k−1

q

q^k n

k

q

q^n−k1 n

k−1

q

n

k

q

, 1.3

see4,7.

Theq-binomial formulas are known, b;q

n 1−b

1−bq

· · ·

1−bq^{n−1 n}

i0

n i

q

q

i 2

−1ⁱbⁱ,

1 b;q

n

1 1−b

1−bq

· · ·

1−bq^{n−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 p^N

−q p^N−1

x0

fx

−q_x

, 1.5

see7,9. Note that

I₋₁ f

lim

q→1I_−q f

Zp

fxdμ₋₁x. 1.6

Forn∈N, writefnx fxn. Then, we have

I₋₁ fn

−1ⁿI₋₁ f

2 n−1

l0

−1^n−1−lfl. 1.7

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

t Zp

e^xytdμ₋₁ y

2t

e^t1e^xt∞

n0

Gnxtⁿ

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:

t^r

Zp

· · ·

Zp

rtimes

e^xx1···xrtdμ−1x1· · ·dμ−1xr

2t e^t1

r

e^xt∞

n0

G^rn xtⁿ n!,

1.9 e^t1

2t r

e^xt∞

n0

G^−rn xtⁿ

n!, 1.10

see7,9. In the special casex0, G^−rn 0 G^−rn , andG^rn 0 G^rn are 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

k0ⁿ_kqq^k2fn−k see5,6,10. Theq-Stirling number of the second kindas defined by Carlitzis given by

S2

n, k;q

q^−k2 k_q!

k j0

−1^jq^j2 k

j

q

k−j_n

q, 1.13

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

S2

n, k;q

q^−k2

k_q!Δ^k_q0ⁿ, 1.14 see6,10.

In this paper, theq-extensions of1.9are considered in several ways. Using theseq- extensions, we derive some interesting identities and relations for Genocchi polynomials and

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,qxtⁿ n! t

Zp

e^xyqtdμ−1 y

^∞

n0

2t

1−q_nⁿ

l0

ⁿ_l−1^lq^lx 1q^l

tⁿ n! 2t

∞ m0

−1^me^mxqt.

2.1

Thus, we obtain the following lemma.

Lemma 2.1. Ifn≥0, then

Gn1,qx n1 2

∞ m0

−1^mmxⁿ_q 2 1−qn

n l0

ⁿ_l−1^lq^lx

1q^l . 2.2

By1.14,

xⁿ_q ⁿ

k0

x k

q

k_q!S2

k, n−k;q q^k2

ⁿ

k0

x_qx−1_q· · ·x−k1_qq^k2−n−k2 n−k_q!Δ^n−k_q 0^k

ⁿ

k0

q^k2−n−k2

n−k_q! Δ^n−k_q 0^k 1 1−q_k

k l0

k l

q

q^2l−1^lq^lx−k1.

2.3

Thus, we have

Gn1,q

n1 ⁿ

k0

q^k2S2

k, n−k;q 1−q_k

k l0

k l

q

q^2l−1^ll

m0

l m

q−1_mGm1,q1−k

m1 , 2.4

and we obtain the following theorem.

Theorem 2.2. Ifn≥0, then Gn1,q

n1 ⁿ

k0

q^k2S2

k, n−k;q 1−q_k

k l0

k l

q

q^2l−1^ll

m0

l m

q−1_mGm1,q1−k

m1 , 2.5

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

Consider aq-extensoin of1.9such thatG^r_0,qx G^r_1,qx · · ·G^r_r−1,qx 0 and G^r_nr,qx

r!^nr_r

Zp

· · ·

Zp

xx1· · ·xrⁿ_qdμ−1x1· · ·dμ−1xr

2^r 1−q_nⁿ

l0

n l

−1^lq^lx 1

1q^l r

2^r ∞ m0

mr−1 m

−1^mmxⁿ_q. 2.6

LetFq^rt, x _∞

n0G^rn,qxtⁿ/n!. Then,

Fq^rt, x 2^rt^r ∞ m0

mr−1 m

−1^me^mxqt. 2.7

In the special case x 0, the numbers G^rn,q0 G^rn,q 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

G^rq t, x Fq^−rt, x 1 2^rt^r

r m0

r m

e^mxqt∞

n0

G^−rn,q xtⁿ

n!. 2.8

By2.8,G^−r_0,q x G^−r_1,q x · · ·G^−r_r−1,qx 0 andr!ⁿrG^r_n−r,qx 1/2^r_r

m0m^rmxⁿ_q. Therefore, we obtain the following theorem.

Theorem 2.3. Forr∈N, and,n≥0, write

2^rt^r ∞ m0

mr−1 m

−1^me^mxqt∞

n0

G^rn,qxtⁿ

n!. 2.9

Then,

G^r_nr,qx r!_nr

r

2^r 1−q_nⁿ

l0

n l

−1^lq^lx 1

1q^l _r

2^r ∞ m0

mr−1 m

−1^mmxⁿ_q,

r!

n r

G^−r_n−r,qx 1 2^r

1−qn

n l0

n l

−1^lq^lx 1q^l_r

1 2^r

r m0

r m

mxⁿ_q.

2.10

The numbersG^−rn,q 0 G^−rn,q 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:

G^h,r_nr,qx r!^nrr

Zp

· · ·

Zp

q^rj1h−jxjxx1· · ·xrⁿ_qdμ₋₁x1· · ·dμ₋₁xr. 2.11

Then,

G^h,r_nr,qx

r!^nr_r 2^r 1−q_nⁿ

l0

ⁿ_l−1^lq^lx −q^h−1l;q⁻¹

r

2^r 1−q_nⁿ

l0

ⁿ_l−1^lq^lx −q^h−rl;q

r

2^r ∞ m0

mr−1 m

q

−1^mq^h−rmxmⁿ_q.

2.12

LetFq^h,rt, x _∞

n0G^h,rn,q xtⁿ/n!. Then, we can readily see that

F^h,rq t, x 2^rt^r ∞ m0

mr−1 m

q

−1^mq^h−rme^xmqt. 2.13

Therefore, we obtain the following theorem.

Theorem 2.4. Forh∈Zandn≥0, let

2^rt^r ∞ m0

mr−1 m

q

−1^mq^h−rme^xmqt∞

n0

G^h,rn,q xtⁿ

n!. 2.14

Then,

G^h,r_nr,qx

r!^nr_r 2^r 1−q_nⁿ

l0

ⁿ_l−1^lq^lx −q^h−rl;q

r

2^r ∞ m0

mr−1 m

q

−1^mq^h−rmxmⁿ_q. 2.15

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

r! n

r

G^h,−r_n−r,qx 1 1−q_nⁿ

l0

ⁿ_l−1^lq^lx

Zp· · ·

Zpq^lx1···xrq^rj1h−jxjdμ−1x1· · ·dμ−1xr. 2.16

By2.16,

r!

n r

G^h,−r_n−r,qx 1 2^r

1−q_nⁿ

l0

n l

−1^lq^lx

−q^h−rl;q

r

1 2^r

r m0

r m

q

q^m2q^h−rmmxⁿ_q.

2.17

LetFq^h,−rt, x _∞

n0G^h,−rn,q xtⁿ/n!. Then, we have

Fq^h,−rt, x 1 2^rt^r

r m0

r m

q

q^m2q^h−rme^mxqt, 2.18

where, G^h,−r_0,q x G^h,−r_1,q x · · · G^h,−r_r−1,qx 0. Therefore, we obtain the following theorem.

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

1 2^rt^r

r m0

r m

q

q^m2q^h−rme^mxqt∞

n0

G^h,−rn,q xtⁿ

n!. 2.19

Then,

r!

n r

G^h,−r_n−r,qx 1 2^r

1−q_nⁿ

l0

n l

−1^lq^lx

−q^h−rl;q

r

1 2^r

r m0

r m

q

q^m2q^h−rmmxⁿ_q,

2.20

where,G^h,−r_0,q x G^h,−r_1,q x · · ·G^h,−r_r−1,qx 0.

Forhr,

G^r,r_nr,qx r!_nr

r

2^r 1−q_nⁿ

l0

_n

l

−1^lq^lx −q^l;q

r

2^r ∞ m0

mr−1 m

q

−1^mxmⁿ_q, 2.21

r!

n r

G^r,−r_n−r,qx 1 2^r

1−q_nⁿ

l0

n l

−1^lq^lx

−q^l;q

r 1

2^r r m0

r m

q

q

m 2

mxⁿ_q. 2.22

It can readily be seen that q^mx2^r

−q^m−r;q

r

Zp

· · ·

Zp

q^rj1m−jxjmxdμ−1x1· · ·dμ−1xr Zp

· · ·

Zp

xx1· · ·xr_q q−1

1_m

q^−rj1jxjdμ₋₁x1· · ·dμ₋₁xr

^m

l0

m l

q−1_l

Zp

· · ·

Zp

xx1· · ·xr^l_qq^−rj1jxjdμ₋₁x1· · ·dμ₋₁xr

^m

l0

m l

q−1lG^0,r_lr,qx r!_lr

r

.

2.23

By2.23,q^mx2^r/−q^m−r;q_{r m}

l0^m_l q−1^lG^0,r_lr,qx/r!_lr

r

. As is known,

I−1 f1

I−1 f

2f0, where f1x fx1. 2.24

It follows from2.24that q^h−1

Zp

· · ·

Zp

x1x1· · ·xrⁿ_qq^rj1h−jxjdμ₋₁x1· · ·dμ₋₁xr −

Zp

· · ·

Zp

xx1· · ·xrⁿ_qq^rj1h−jxjdμ−1x1· · ·dμ−1xr

2

Zp

· · ·

Zp

xx2· · ·xrⁿ_qq^{r−1j1h−1−jxj1}dμ−1x2· · ·dμ−1xr.

2.25

By2.25,

q^h−1G^h,r_nr,qx1

nr G^h,r_nr,qx

nr 2G^h−1,r−1_nr−1,q x. 2.26

A simple manipulation shows that q^x

Zp

· · ·

Zp

xx1· · ·xrⁿ_qq^rj1h−j1xjdμ−1x1· · ·dμ−1xr

q−1

Zp

· · ·

Zp

xx1· · ·xrⁿ¹_q q^rj1h−jxjdμ−1x1· · ·dμ−1xr

Zp

· · ·

Zp

xx1· · ·xrⁿ_qq^rj1h−jxjdμ−1x1· · ·dμ−1xr.

2.27

By2.27,q^xG^h1,r_nr,q x/n1 q−1G^h,r_nr1,qx/nr1 G^h,r_nr,qx/n1.

Therefore, we obtain the following proposition.

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

q^h−1G^h,r_nr,qx1

nr G^h,r_nr,qx

nr 2G^h−1,r−1_nr−1,q x, q^xG^h1,r_nr,q x

n1

q−1G^h,r_nr1,qx

nr1 G^h,r_nr,qx n1

2.28

hold. Moreover,q^mx2^r/−q^m−r;q_{r m}

l0^m_lq−1^lG^0,r_lr,qx/r!_lr

r

.

By2.21, G^r,r_nr,q−1r−x

r!^nrr 2^r 1−q⁻¹_nⁿ

l0

ⁿ_l−1^lq^−lr−x

−q^−l;q⁻¹

r

−1ⁿq^nr2 2^r 1−q_nⁿ

l0

ⁿ_l−1^lq^x −q^l;q

r

−1ⁿq^nr2G^r,r_nr,qx r!^nrr .

2.29

Hence,

Zp

· · ·

Zp

r−xx1· · ·xrⁿ_q−1q^{−rj1r−jxj}dμ₋₁x1· · ·dμ₋₁xr

−1ⁿq^nr2

Zp

· · ·

Zp

xx1· · ·xrⁿ_qq^rj1r−jxjdμ−1x1· · ·dμ−1xr.

2.30

Forhr,G^r,r_nr,q−10 −1ⁿq^nr2G^r,r_nr,qr. It also follows from2.26that

q^r−1G^r,r_nr,qx1

nr G^r,r_nr,qx

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;q

z^k, 2.32

see6,9,

q^m2 r

m

q

q^m2r_q· · ·r−m1_q m_q! 1

m_q! m−1

k0

r_q−k_q

. 2.33

It can readily be seen that n−1

k0

z−k_q zⁿ

n−1 k0

1− k_q

z

ⁿ

k0

S1

n−1, k;q

−1^kz^n−k. 2.34

By2.33and2.34, m−1

k0

r_q−k_q ^m

k0

S1

m−1, k;q

−1^kr^m−k_q . 2.35

Formulas2.22and2.35imply the following assertion.

Proposition 2.7. Forr∈Nandn∈Z, r!

n r

G^r,−r_n−r,qx 1 2^rm_q!

r m0

m k0

S1

m−1, k;q

−1^kr^m−k_q mxⁿ_q. 2.36

The generalized Genocchi numbers and polynomials of N ¨orlund type are defined by 2^rt^r

e^w1t1e^w2t1· · ·e^wrt1e^xt∞

n0

G^rn x|w1, . . . , wrtⁿ

n!, 2.37

andG^rn w1, . . . , wr G^rn 0|w1, . . . , wr. We can now also define aq-extension of2.37as follows. Forw1, . . . , wr ∈Zpandδ1, . . . , δr ∈Z, write

G^r_nr,qx|w1, . . . , wr;δ1, . . . , δr

r!^nrr

Zp

· · ·

Zp

x1w1· · ·xrwrxⁿ_qdμ_−q^δ1x1· · ·dμ_−qδrxr, 2.38

andG^r_nr,qw1, . . . , wr;δ1, . . . , δr G^r_nr,q0|w1, . . . , wr;δ1, . . . , δr. Thus, G^r_nr,qx|w1, . . . , wr;δ1, . . . , δr

r!^nrr 2_q^δ1· · ·2_qδr

1−q_n ⁿ

l0

ⁿ_l−1^lq^lx 1q^δ1lw1

· · ·

1q^δrlwr. 2.39 Anotherq-extension of N ¨orlund type generalized Genocchi numbers and polynomials is also of interest, namely,

G^∗r_nr,qx|w1, . . . , wr;δ1, . . . , δr r!^nr_r

Zp

· · ·

Zp

x1w1· · ·xrwrxⁿ_qq^{δ1x1···δrxr}dμ−1x1· · ·dμ−1xr,

2.40

andG^∗r_nr,qw1, . . . , wr;δ1, . . . , δr G^∗r_nr,q0|w1, . . . , wr;δ1, . . . , δr. By2.40,

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

r!^nr_r 2^r

1−qn

n l0

ⁿ_l−1^lq^lx 1q^δ1lw1

· · ·

1q^δrlwr. 2.41

3. Further Remarks

Forh0, consider the following polynomialsG^0,r_nr,qx/r!^nr_r andr!ⁿrG^0,−r_nr,qx:

G^0,r_nr,qx r!_nr

r

Zp

· · ·

Zp

xx1· · ·xrⁿ_qq^−rj1jxjdμ−1x1· · ·dμ−1xr,

r!

n r

G^0,−r_n−r,qx 1 1−q_nⁿ

l0

_n

l

−1^lq^lx

Zp· · ·

Zpq^lx1···xrq^−rj1jxjdμ−1x1· · ·dμ−1xr.

3.1

Then,

G^0,r_nr,qx r!_nr

r

2^r 1−qn

n l0

_n

l

−1^lq^lx −q^l−r;q

r

2^r ∞ m0

mr−1 m

q

q^−rm−1^mxmⁿ_q

r!

n r

G^0,−r_n−r,qx 1 2^r

1−q_nⁿ

l0

n l

−1^lq^lx

−q^l−r;q

r 1

2^r r m0

r m

q

q

m 2

q^−rmmxⁿ_q. 3.2

LetFq^0,rt, x _∞

n0G^0,rn,q xtⁿ/n!and letFq^0,−rt, x _∞

n0G^0,−rn,q xtⁿ/n!. Then,

Fq^0,rt, x 2^rt^r ∞ m0

mr−1 m

q

q^−rm−1^me^xmqt,

Fq^0,−rt, x 1 2^rt^r

r m0

r m

q

q

m 2

q^−rme^mxqt.

3.3

Consider the following polynomials:

G^h,1_n1,qx n1

Zp

q^x1h−1xx1ⁿ_qdμ₋₁x1

2

1−q_nⁿ

l0

ⁿ_l−1^lq^lx

1q^lh−1 . 3.4

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

q^x

Zp

xx1ⁿ_qq^x1h−1dμ₋₁x1

q−1

Zp

xx1ⁿ¹_q q^x1h−2dμ₋₁x1 Zp

xx1ⁿ_qq^x1h−2dμ₋₁x1.

3.5

By3.5,q^xG^h,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

xx1ⁿ_qq^x1h−1dμ₋₁x1

n j0

n j

x^n−jq q^jx

Zp

x1^jqq^x1h−1dμ₋₁x1. 3.6

By3.6,G^h,1_n1,qx/n1 _n

j0_n

j

x^n−jq q^jxG^h,1_j1,q/j1. Using2.24, we can also prove that

Zp

xx11ⁿ_qq^x11h−1dμ₋₁x1 Zp

xx1ⁿ_qq^x1h−1dμ₋₁x1 2xⁿ_q. 3.7

Thus,q^h−1G^h,1_n1,qx/n1 G^h,1_n1,qx/n1 2xⁿ_q. Forx0, we haveq^h−1G^h,1_n1,q1/

n1 G^h,1_n1,q/n1 2δn,0, whereδn,0is the Kronecker delta.

It is easy to see thatG^h,1_1,q

Zpq^x1h−1dμ−1x1 2/1q^h−1 2/2_qh−1. By3.4,

G^h,1_n1,q−11−x

n1

Zp

1−xx1ⁿ_q−1q^−x1h−1dμ₋₁x1

−1ⁿq^nh−1 2 1−q_nⁿ

l0

ⁿ_l−1^lq^lx 1q^lh−1 −1ⁿq^nh−1G^h,1_n1,qx

n1 .

3.8

In particular, if x 1, then G^h,1_n1,q−10/n 1 −1ⁿq^nh−1G^h,1_n1,q1/n 1

−1ⁿ⁻¹qⁿG^h,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 6,14,20.

References

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

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Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340–348, 2006.

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