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-ing q-deformed fermionic integral on Zp, 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, and Cp will denote the ring of rational integers,
the set of positive integers, the ring ofp-adic integers, the field ofp-adic rational numbers, and
valuation ofCp with |p|p p−vpp p−1. Here,q is variously considered as an indeterminate,
a complex numberq ∈ C, or p-adic numberq ∈Cp. Ifq ∈Cp, then we assume that|q−1|p <
p−1/p−1, so that qx expxlogq for |x|
p ≤ 1. If q ∈ C, then we assume that |q| < 1 cf.
1,3,4,9.
We use the following notations:
xq 1−q
x
1−q , x−q
1−−qx
1q . 1.1
Note that limq→1xqx.
Let UDZpbe the set of uniformly differentiable functions onZp. Letf ∈ UDZp,Cp
{f|f :Zp →Cpis uniformly differentiable function}. Forf∈UDZp,Cp, let
1
pN q
pN−1
x0
fxqx
pN−1
x0
fxμqadpNZp
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
fxdμqx lim
N→∞
1
pN q
pN−1
x0
fxqx, 1.3
where
μqadpNZp
qa dpN
q
, N ∈Z. 1.4
Note that iff ∈UDZp,Cp, then
Zp
fxdμqx
p
≤pf1, 1.5
where
f1sup f0p,sup x/y
fxx−−fyy
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 on Zp, is
de-fined by
I−qf qlim→−qIqf
Zp
where
μ−qadpNZp
−qa dpN
−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 and q-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
where Cpn {w | wpn 1}is the cyclic group of orderpn. Forw ∈ Tp,φ
w : Zp → Cp is the
locally constant functionx→wxcf.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,En,whx, q, as
follows:
Fw,qht, x Fw,qhtetx 2e tx
wqhet1
∞
n0
En,whx, qt n
n!. 1.12
Note that ifw→1, thenEn,whq→Enhqand
Fw,qht−→Fqht
2
qhet1 1.13
cf.7. Ifq→1, then
Fqht−→Ft
2
et1
∞
n1
Ent n
n!, 1.14
whereEnis usual Euler numberscf.3,8,10.
Forx0, we have
Fqht
2
wqhet1
∞
n0
En,whqt n
Theorem 1.115Witt formula. Forh∈Z,q∈Cpwith|1−q|p < p−1/p−1,
Zp
qhxwnxndμ−1x En,whq, 1.16
Zp
qhyxyndμ−1y En,whx, 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 wqh v j1xj
exp
t
v
j1
xj
v
j1
dμ−1
xj
∞ n0
En,wh,vqt n
n!, 2.1
wherevj1dμ−1xj dμ−1x1dμ−1x2· · ·dμ−1xv. By using the above equation, we easily
have
∞
n0 Zp
· · · Zp wqh v j1xj
v
j1
xj
n v
j1
dμ−1xj
tn
n!
∞
n0
En,wh,vqt n
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,wh,vq
Zp · · · Zp wqh v j1xj
v
j1
xj
n v
j1
dμ−1xj
. 2.3
By2.1, twistedh, q-Euler numbers of higher-order,En,wh,vq, are defined by means of
the following generating function:
2
wqhet1
v
∞ n0
En,wh,vqt n
n!. 2.4
Observe that forv 1, the above equation reduces to1.15:
Zp · · · Zp v-times wqh v j1xj
exp
tz
v
j1
txj
v
j1
dμ−1xj
∞ n0
En,wh,vz, qt n
n!. 2.5
By using Taylor series of exptxin the above equation, we have
∞
n0 Zp
· · · Zp wqh v j1xj
z
v
j1
xj
n v
j1
dμ−1
xj tn n! ∞
n0
En,wh,vz, qt n
n!. 2.6
Theorem 2.2Witt-type formula. Forz∈Cpand positive integersn,v, andh∈Z, then
En,wh,vz, q
Zp
· · ·
Zp
qhw
v j1xj
z
v
j1
xj
n v
j1
dμ−1
xj
. 2.7
By2.1,h, q-Euler polynomials of higher-order,En,qh,vz, are defined by means of the
following generating function:
Fq,wh,vz, t etz
2
wqhet1
v ∞
n0
En,wh,vz, qt n
n!. 2.8
Note that whenv1, then we have1.12; whenq→1 andw→1, then we have
Fvz, t etz
2
et1
v
∞ n0
Envzt n
n!, 2.9
whereEnvzdenote classical higher-order Euler polynomialscf.10.
Theorem 2.3. Forz∈Cpand positive integersn, v,andh∈Z, then
En,wh,vz, q n
l0
n
l
zn−lEl,wh,vq. 2.10
Proof. By using binomial expansion in2.7, we have
En,wh,vz, q n
l0
n
l
zn−l
Zp
· · ·
Zp
qhw
v j1xj
v
j1
xj
l v
j1
dμ−1xj
. 2.11
By2.3in the above, we arrive at the desired result.
Remark 2.4. Ifw→ 1, thenEn,wh,vq→ Enh,vq cf.11. Ifq →1,v 1 , thenEn,wh,vq→ En,
whereEn,wv is 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 of h, 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
xala, 3.1
wherel1,l2n,...,lvare the multinomial coefficients, which are defined by
n
Theorem 3.2. For positive integersn,v, then
En,wh,vq
l1,l2,...,lv≥0
l1l2···lvn
n l1, l2, . . . , lv
v
j1
Eljh,wq, 3.2
wherel1,l2n,...,lvis the multinomial coefficient.
Proof. By usingTheorem 3.1in2.3, we have
En,wh,vq
l1,l2,...,lv≥0
l1l2···lvn
n l1, l2, . . . , lv
v
j1
Zp
wqhxjxljjdμ−1xj
. 3.3
By1.16in the above, we obtain the desired result.
By substituting3.2into2.10, we have the following corollary.
Corollary 3.3. Forz∈Cp and positive integersn,v, then
En,wh,vz, q n
m0
l1,l2,...,lv≥0
l1l2···lvm
n m
m l1, l2, . . . , lv
zn−m
v
j1
Elj,whq. 3.4
Complete sum of products of the twisted h, q-Euler polynomials is given by the
fol-lowing theorem.
Theorem 3.4. Fory1, y2, . . . , yv ∈Cpand positive integersn, v,then
En,wh,v
y1y2· · · yv, q
l1,l2,...,lv≥0
l1l2···lvn
n l1, l2, . . . , lv
v
j1
Elj,whyj, q
. 3.5
Proof. By substitutingzy1y2· · · yvinto2.7, we have
En,wh,v
y1y2· · · yv, q
Zp
· · ·
Zp
wqh
v j1xj
v
j1
yjxj
n v
j1
dμ−1xj
. 3.6
By usingTheorem 3.1in the above, and after some elementary calculations, we get
En,wh,v
y1y2· · · yv, q
l1,l2,...,lv≥0
l1l2···lvn
n l1, l2, . . . , lv
v
j1
Zp
wqhxjy jxj
lj
dμ−1xj
. 3.7
Remark 3.5. If we take y1 y2 · · · yv 0 in Theorem 3.4, then Theorem 3.4reduces to Theorem 3.2. Substitutingq→1 andw→1 into3.5, we obtain the following relation:
Env
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. Huang 20found complete sums of products of Bernoulli
polynomi-als. Kim 13defined Carlitz’s q-Bernoulli number of higher order using an integral by the
q-analogueμqof the ordinaryp-adic invariant measure. He gave a different proof of complete
sums of products of higher order q-Bernoulli polynomials. In 21, 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
En,wh,vzy, q n
l0
n
l
El,wh,vy, qzn−l. 3.9
Proof. Assume
En,wh,vzy, q
Ewh,vq zy
n
n l0
n
l
El,wh,vqyzn−l
3.10
with usual convention of symbolically replacingElwh,vbyEl,wh,vq. By using2.10in the above,
we have
En,wh,vzy, q n
m0
n m
Em,wh,vy, qzn−m. 3.11
Thus the proof is completed.
From Theorems3.4and3.6, after some elementary calculations, we arrive at the
follow-ing interestfollow-ing result.
Corollary 3.7. Letn∈Z. Then
n
m0
n m
Em,wh,v
y1, q
yn2−m
l1,l2≥0
l1l2n
n l1, l2
El1h,wy1, q
Bl2h,wy2, q
. 3.12
Acknowledgments
References
1T. Kim, “q-Volkenborn integration,”Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.
2T. Kim, “A new approach toq-zeta function,”Advanced Studies in Contemporary Mathematics, vol. 11, no. 2, pp. 157–162, 2005.
3T. Kim, “On the analogs of Euler numbers and polynomials associated withp-adicq-integral onZpat
q−1,”Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 779–792, 2007.
4T. Kim, “On the q-extension of Euler and Genocchi numbers,”Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1458–1465, 2007.
5T. Kim, “q-Euler numbers and polynomials associated withp-adicq-integrals,”Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007.
6T. Kim, “An invariantp-adicq-integral onZp,”Applied Mathematics Letters, vol. 21, no. 2, pp. 105–108, 2008.
7H. Ozden and Y. Simsek, “A new extension of q-Euler numbers and polynomials related to their interpolation functions,” to appear inApplied Mathematics Letters.
8Y. Simsek, “q-analogue of twistedI-series andq-twisted Euler numbers,” Journal of Number Theory, vol. 110, no. 2, pp. 267–278, 2005.
9Y. Simsek, “Twistedh, q-Bernoulli numbers and polynomials related to twistedh, q-zeta function andL-function,”Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 790–804, 2006. 10H. M. Srivastava, T. Kim, and Y. Simsek, “q-Bernoulli numbers and polynomials associated with
mul-tipleq-zeta functions and basicL-series,”Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 241–268, 2005.
11Y. Simsek, “Complete sum of products ofh, q-extension of the Euler Polynomials and numbers,” preprint, 2007,http://arxiv.org/abs/0707.2849v1.
12T. Kim and S.-H. Rim, “On the twistedq-Euler numbers and polynomials associated with basicq−l -functions,”Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 738–744, 2007.
13T. Kim, “Sums of products ofq-Bernoulli numbers,”Archiv der Mathematik, vol. 76, no. 3, pp. 190–195, 2001.
14Y. Simsek, V. Kurt, and D. Kim, “New approach to the complete sum of products of the twistedh, q -Bernoulli numbers and polynomials,”Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 44– 56, 2007.
15H. Ozden, I. N. Cangul, and Y. Simsek, “Generating functions of theh, q-extension of Euler polyno-mials and numbers,” to appear inActa Mathematica Hungarica.
16T. Kim, L. C. Jang, S.-H. Rim, and H.-K. Pak, “On the twistedq-zeta functions andq-Bernoulli poly-nomials,”Far East Journal of Applied Mathematics, vol. 13, no. 1, pp. 13–21, 2003.
17L. C. Jang, H. K. Pak, S.-H. Rim, and D.-W. Park, “A note on analogue of Euler and Bernoulli numbers,” JP Journal of Algebra, Number Theory and Applications, vol. 3, no. 3, pp. 461–469, 2003.
18L. Comtet,Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel, Dordrecht, The Netherlands, 1974.
19R. L. Graham, D. E. Knuth, and O. Patashnik,Concrete Mathematics. A Foundation for Computer Science, Addison-Wesley, Reading, Mass, USA, 1989.
20I.-C. Huang and S.-Y. Huang, “Bernoulli numbers and polynomials via residues,”Journal of Number Theory, vol. 76, no. 2, pp. 178–193, 1999.
