Remarks on Sum of Products of h, q -Twisted Euler Polynomials and Numbers

Journal of Inequalities and Applications Volume 2008, Article ID 816129,8pages doi:10.1155/2008/816129

Research Article

Remarks on Sum of Products of h, q -Twisted Euler Polynomials and Numbers

Hacer Ozden,¹Ismail Naci Cangul,¹and Yilmaz Simsek²

1Department of Mathematics, Faculty of Arts and Science, University of Uludag, 16059 Bursa, Turkey

2Department of Mathematics, Faculty of Arts and Science, University of Akdeniz, 07058 Antalya, Turkey

Correspondence should be addressed to Hacer Ozden,[email protected] Received 29 March 2007; Accepted 16 October 2007

Recommended by Panayiotis D. Siafarikas

The main purpose of this paper is to construct generating functions of higher-order twistedh, q- extension of Euler polynomials and numbers, by usingp-adic,q-deformed fermionic integral onZp. By applying these generating functions, we prove complete sums of products of the twistedh, q- extension of Euler polynomials and numbers. We also define some identities involving twisted h, q-extension of Euler polynomials and numbers.

Copyrightq2008 Hacer Ozden 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.

1. Introduction, definitions, and notations

Higher-order twisted Bernoulli and Euler numbers and polynomials were studied by many au- thorssee for details1–10. In1,3, Kim constructedp-adic,q-Volkenborn integral identities.

He provedp-adic,q-integral representation ofq-Euler and Bernoulli numbers and polynomi- als. In11, the second author constructed a new approach to the complete sums of products ofh, q-extension of higher-order Euler polynomials and numbers. Kim and Rim12, by us- ingq-deformed fermionic integral onZp, defined twisted generating functions of theq-Euler numbers and polynomials, respectively. By using these functions, they also constructed inter- polation functions of these numbers and polynomials.

By the same motivation of the above studies, in this paper, we construct a new approach to the complete sums of products of twistedh, q-extension of Euler polynomials and num- bers.

Throughout this paper,Z,Z,Zp,Qp, andCpwill denote the ring of rational integers, the set of positive integers, the ring ofp-adic integers, the field ofp-adic rational numbers, and the completion of the algebraic closure ofQp, respectively. Letvpbe the normalized exponential

valuation ofCpwith|p|p p^−vpp p⁻¹. Here,qis variously considered as an indeterminate, a complex numberq ∈C, orp-adic numberq ∈Cp. Ifq ∈Cp, then we assume that|q−1|p <

p^−1/p−1, so thatq^x expxlogqfor |x|p ≤ 1. Ifq ∈ C, then we assume that|q| < 1 cf.

1,3,4,9.

We use the following notations:

x_q 1−q^x

1−q , x_−q 1−−q^x

1q . 1.1

Note that lim_q→1x_qx.

Let UDZpbe the set of uniformly differentiable functions onZp. Letf ∈ UDZp,Cp {f|f:Zp→Cpis uniformly differentiable function}. Forf∈UDZp,Cp, let

1 p^N_q

p^N−1

x0fxq^x

p^N−1

x0fxμ_q

adp^NZp

1.2

representing theq-analogue of the Riemann sums forf. The integral offonZpis defined as the limitN→ ∞of the above sums when it exists. Thus, Kim1,3defined thep-adic invariant q-integral onZpas follows:

Iqf

Zp

fxdμ_qx lim

N→∞

1 p^N

q p^N−1

x0

fxq^x, 1.3

where

μ_q

adp^NZp

q^a dp^N

q

, N∈Z. 1.4

Note that iff∈UDZp,Cp, then

Zp

fxdμ_qx

_p≤pf₁, 1.5

where

f₁sup f0

p,sup

x/y

fx−fy x−y

_p

cf.3. 1.6

The bosonic integral was considered from a physical point of view to the bosonic limitq→1, I1f limq→1Iqf cf.1,3,4,12. By using theq-bosonic integral onZp, not only generating functions of the Bernoulli numbers and polynomials are constructed but also Witt-type formula of these numbers and polynomials are definedcf. for detail1,9,10,13,14.

The fermionic integral, which is called theq-deformed fermionic integral onZp, is de- fined by

I−qf lim

q→−qIqf

Zp

fxdμ_−qx, 1.7

where

μ_−q

adp^NZp

−q^a

dp^N_−q, N∈Z cf.3,4,6,12. 1.8 In view of the notationI−1is written symbolically by

I−1f lim

q→−1Iqf. 1.9

By usingq-deformed fermionic integral onZp, generating functions of the Euler numbers and polynomials, Genocchi numbers and polynomials, and Frobenius-Euler numbers and polyno- mials are constructedcf. for detail1,3,6–8,10–12,15.

The main motivation of this paper is to construct generating functions of higher-order twistedh, q-extension of Euler polynomials and numbers by usingq-deformed fernionic in- tegral onZp. Moreover, by this integral, we also define Witt-type formula of the higher-order twistedh, q-extension of Euler polynomials and numbers. By applying these generating func- tions andq-deformed fernionic integral, we obtain complete sums of products of the twisted h, q-extension of Euler polynomials and numbers as well.

The twistedh, q-Bernoulli and Euler numbers and polynomials have been studied by several authorscf.5,8,9,11,15–17.

In3,6, Kim defined the following integral equation: forf1x fx1, I−1

f1

I−1f 2f0. 1.10 Let

Tp

n≥1

Cpⁿ lim

n→∞Cpⁿ, 1.11

whereCpⁿ {w | w^pn 1}is the cyclic group of orderpⁿ. Forw ∈Tp,φ_w : Zp → Cpis the locally constant functionx→w^xcf.9,14,16.

Ozden and Simsek7defined newh, q-extension of Euler numbers and polynomials.

In15, Ozden et al. also defined twistedh, q-extension of Euler polynomials,E^hn,wx, q, as follows:

Fw,q^ht, x Fw,q^hte^tx 2e^tx

wq^he^t1 ^∞

n0

En,w^hx, qtⁿ

n!. 1.12

Note that ifw→1, thenEn,w^hq→E^hn qand

Fw,q^ht−→Fq^ht 2

q^he^t1 1.13

cf.7. Ifq→1, then

Fq^ht−→Ft 2

e^t1 ^∞

n1

Entⁿ

n!, 1.14

whereEnis usual Euler numberscf.3,8,10.

Forx0, we have

Fq^ht 2

wq^he^t1 ^∞

n0E^hn,wqtⁿ

n! cf.7. 1.15

Theorem 1.115Witt formula. Forh∈Z,q∈Cpwith|1−q|p< p^−1/p−1,

Zp

q^hxwⁿxⁿdμ₋₁x E^hn,wq, 1.16

Zp

q^hyxyⁿdμ₋₁y E^hn,wx, q. 1.17

2. Higher-order twistedh, q-Euler polynomials and numbers

Here, we study on higher-order twistedh, q-Euler polynomials and numbers and complete sums of products of these polynomials and numbers, our method is similar to that of11. For constructions of them, we use multiple theq-deformed fermionic integral onZp:

Zp

· · ·

Zp

v-times

wq^hv_j1xj

exp

t v

j1

xj

_v

j1

dμ₋₁ xj

^∞

n0

E^h,vn,w qtⁿ

n!, 2.1

where_v

j1dμ₋₁xj dμ₋₁x1dμ₋₁x2· · ·dμ₋₁xv. By using the above equation, we easily have

∞

n0 Zp

· · ·

Zp

wq^hv_j1xj

_v

j1

xj

_{n v}

j1

dμ₋₁ xj

tⁿ n! ^∞

n0

En,w^h,vqtⁿ

n!. 2.2

By comparing coefficients oftⁿ/n! in the above equation, we have the following theorem.

Theorem 2.1. For positive integersn,v, andh∈Z, then En,w^h,vq

Zp

· · ·

Zp

wq^hv_j1xj

_v

j1

xj

_{n v}

j1

dμ₋₁ xj

. 2.3

By2.1, twistedh, q-Euler numbers of higher-order,E^h,vn,w q, are defined by means of the following generating function:

2 wq^he^t1

v

^∞

n0

E^h,vn,w qtⁿ

n!. 2.4

Observe that forv1, the above equation reduces to1.15:

Zp

· · ·

Zp

v-times

wq^hv_j1xj

exp

tz^v

j1

txj

_v

j1

dμ₋₁ xj

^∞

n0

E^h,vn,w z, qtⁿ

n!. 2.5

By using Taylor series of exptxin the above equation, we have ∞

n0 Zp

· · ·

Zp

wq^hv_j1xj

z^v

j1

xj

_{n v}

j1

dμ₋₁ xj

tⁿ n!^∞

n0

E^h,vn,w z, qtⁿ

n!. 2.6 By comparing coefficients oftⁿ/n! in the above equation, we arrive at the following theorem.

Theorem 2.2Witt-type formula. Forz∈Cpand positive integersn,v, andh∈Z, then

E^h,vn,w z, q

Zp

· · ·

Zp

q^hw^v_j1xj

z^v

j1

xj

_{n v}

j1

dμ₋₁ xj

. 2.7

By2.1,h, q-Euler polynomials of higher-order,E^h,vn,q z, are defined by means of the following generating function:

Fq,w^h,vz, t e^tz 2

wq^he^t1 _v

^∞

n0

E^h,vn,w z, qtⁿ

n!. 2.8

Note that whenv1, then we have1.12; whenq→1 andw→1, then we have F^vz, t e^tz

2 e^t1

_v ^∞

n0E^vn ztⁿ

n!, 2.9

whereE^vn zdenote classical higher-order Euler polynomialscf.10.

Theorem 2.3. Forz∈Cpand positive integersn, v,andh∈Z, then

E^h,vn,w z, q ⁿ

l0

n l

z^n−lE^h,v_l,w q. 2.10

Proof. By using binomial expansion in2.7, we have

En,w^h,vz, q ⁿ

l0

n l

z^n−l

Zp

· · ·

Zp

q^hw^v_j1xj

_v

j1

xj

_{l v}

j1

dμ₋₁ xj

. 2.11

By2.3in the above, we arrive at the desired result.

Remark 2.4. Ifw→1, thenEn,w^h,vq→E^h,vn q cf.11. Ifq→1,v 1 , thenE^h,vn,w q→En, whereE^vn,wis usual twisted Euler numberscf.10.

3. The complete sums of products ofh, q-extension of Euler polynomials and numbers

In this section, we prove main theorems related to the complete sums of products ofh, q- extension of Euler polynomials and numbers. Firstly, we need the multinomial theorem, which is given as followscf.18,19.

Theorem 3.1multinomial theorem. Let _v

j1

xj

_n

l1,l2,...,lv≥0 l1l2···lvn

n l1, l2, . . . , lv

_v

a1

x^la^a, 3.1

where_l1,l2ⁿ_,...,lvare the multinomial coefficients, which are defined by_l1,l2ⁿ_,...,lv n!/l1!l2!· · ·lv!.

Theorem 3.2. For positive integersn,v, then

En,w^h,vq

l1,l2,...,lv≥0 l1l2···lvn

n l1, l2, . . . , lv

_v

j1

E^h_lj,wq, 3.2

where_l1,l2ⁿ_,...,lvis the multinomial coefficient.

Proof. By usingTheorem 3.1in2.3, we have

E^h,vn,w q

l1,l2,...,lv≥0 l1l2···lvn

n l1, l2, . . . , lv

_v

j1

Zp

wq^hxj

x^l_j^jdμ₋₁ xj

. 3.3

By1.16in the above, we obtain the desired result.

By substituting3.2into2.10, we have the following corollary.

Corollary 3.3. Forz∈Cpand positive integersn,v, then

E^h,vn,w z, q ⁿ

m0

l1,l2,...,lv≥0 l1l2···lvm

n m

m l1, l2, . . . , lv

z^n−m

v j1

E^h_lj,wq. 3.4

Complete sum of products of the twistedh, q-Euler polynomials is given by the fol- lowing theorem.

Theorem 3.4. Fory1, y2, . . . , yv∈Cpand positive integersn, v,then

En,w^h,v

y1y2· · · yv, q

l1,l2,...,lv≥0 l1l2···lvn

n l1, l2, . . . , lv

_v

j1

E^h_lj,w yj, q

. 3.5

Proof. By substitutingzy1y2· · · yvinto2.7, we have

E^h,vn,w

y1y2· · ·yv, q

Zp

· · ·

Zp

wq^hv_j1xj

_v

j1

yjxj

_{n v}

j1

dμ₋₁ xj

. 3.6

By usingTheorem 3.1in the above, and after some elementary calculations, we get E^h,vn,w

y1y2· · ·yv, q

l1,l2,...,lv≥0 l1l2···lvn

n l1, l2, . . . , lv

_v

j1

Zp

wq^hxj yjxj

_lj dμ₋₁

xj

. 3.7

By substituting1.17into the above, we arrive at the desired result.

Remark 3.5. If we takey1 y2 · · · yv 0 inTheorem 3.4, thenTheorem 3.4 reduces to Theorem 3.2. Substitutingq→1 andw→1 into3.5, we obtain the following relation:

E^vn

y1y2· · ·yv

l1,l2,...,lv≥0 l1l2···lvm

m l1, l2, . . . , lv

_v

j1

Elj

yj

cf.11. 3.8

I.-C. Huang and S.-Y. Huang20found complete sums of products of Bernoulli polynomi- als. Kim13defined Carlitz’sq-Bernoulli number of higher order using an integral by the q-analogueμ_qof the ordinaryp-adic invariant measure. He gave a different proof of complete sums of products of higher orderq-Bernoulli polynomials. In21, Jang et al. gave complete sums of products of Bernoulli polynomials and Frobenious Euler polynomials. In14, Simsek et al. gave complete sums of products ofh, q-Bernoulli polynomials and numbers.

Theorem 3.6. Letn∈Z. Then

E^h,vn,w zy, q ⁿ

l0

n l

E_l,w^h,vy, qz^n−l. 3.9

Proof. Assume

E^h,vn,w zy, q

Ew^h,vq zy_n ⁿ

l0

n l

E^h,v_l,w qyz^n−l 3.10

with usual convention of symbolically replacingE^lh,vw byE_l,w^h,vq. By using2.10in the above, we have

E^h,vn,w zy, q ⁿ

m0

n m

E^h,vm,wy, qz^n−m. 3.11

Thus the proof is completed.

From Theorems3.4and3.6, after some elementary calculations, we arrive at the follow- ing interesting result.

Corollary 3.7. Letn∈Z. Then n

m0

n m

E^h,vm,w

y1, q

y₂^n−m

l1,l2≥0 l1l2n

n l1, l2

E^h_l1,w

y1, q B_l^h_2,w

y2, q

. 3.12

Acknowledgments

The first and second authors are supported by the research fund of Uludag University Project no. F-2006/40 and F-2008/31. The third author is supported by the research fund of Akdeniz University.

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