A note on the modified q-Genocchi numbers and polynomials with weight (α,β) and their interpolation function at negative integers

0.14 MB·English

by  Serkan Araci

A NOTE ON THE MODIFIED q-GENOCCHI NUMBERS AND
POLYNOMIALS WITH WEIGHT (α,β) AND THEIR
INTERPOLATION FUNCTION AT NEGATIVE INTEGERS
2
1
0 SERKANARACI,MEHMETAC¸IKGO¨Z,FENGQI,ANDHASSANJOLANY
2

n
Abstract. Thepurposeofthispaperconcernstoestablishmodifiedq-Genocchi
a
numbers and polynomials with weight (α,β). In this paper we investigate
J
special generalized q-Genocchi polynomials and weapply the method of gen-
7
erating function, which are exploited to derive further classes of q-Genocchi
2
polynomials anddevelopq-Genocchi numbers andpolynomials. Byusingthe

Laplace-Mellintransformationintegral,wedefineq-Zetafunctionwithweight
]
O (α,β)andbypresentingalinkbetweenq-Zetafunctionwithweight(α,β)and
q-Genocchinumberswithweight(α,β)weobtainaninterpolationformulafor
C the q-Genocchi numbers and polynomials with weight (α,β). Also we derive
. distribution formula (Multiplication Theorem) and Witt’s type formula for
h
modifiedq-Genocchinumbersandpolynomialswithweight(α,β)whichyields
t
a a deeper insight into the effectiveness of this type of generalizations for q-
m Genocchi numbers and polynomials. Our new generating function possess a
numberofinterestingpropertieswhichwestateinthispaper.
[

2
v
2 1. Introduction, Definitions and Notations
0
9
5 Recently, q-calculus has served as a bridge between mathematics and physics.
. Therefore,thereisasignificantincreaseofactivityintheareaoftheq-calculusdue
2
1 to applications of the q-calculus in mathematics, statistics and physics. The ma-
1 jority of scientists in the world who use q-calculus today are physicists. q-Calculus
1 is a generalization of many subjects, like hypergeometric series, generating func-
:
v tions,complexanalysis,andparticlephysics. Inshort,q-calculusisquiteapopular
i subject today. One of Important Branch of q-calculus in number theory is q-type
X
ofspecialgeneratingfunctions, for instance q-Bernoullinumbers, q-Eulernumbers,
r
and q-Genocchi numbers, here we introduce a new class of q-type generating func-
a
tion. We introduceq-Genocchinumbers with weight(α,β). When we define a new
class of generating functions like, q-Genocchi numbers with weight (α,β), then we
facetowiththisquestionthat“canwedefine anew q-Zetatypefunctioninrelated
ofthis new classofq-type generatingfunction?”. We givea positiveanswerforour
newclassofnumbersandpolynomials. Morepreciselyweshowthatourq-typegen-
erating function is generalization of the Hurwitz Zeta function. Historically many
authors have tried to give q-analogues of the Riemann Zeta function ζ(s), and its
related functions. By just following the method of Kaneko et al. [M. Kaneko, N.
KurokawaandM.Wakayama,AvariationofEuler’sapproachtotheRiemannZeta
Date:December12,2011.
2000 Mathematics Subject Classification. Primary46A15,Secondary41A65.
Key words and phrases. Genocchi numbers and polynomials, q-Genocchi numbers and poly-
nomials,q-Genocchi numbersandpolynomialswithweightα.
1
2 SERKANARACI,MEHMETAC¸IKGO¨Z,FENGQI,ANDHASSANJOLANY
function, Kyushu J. Math. 57 (2003), 175–192],who mainly used Euler-Maclaurin
summation formula to present and investigate a q-analogue of the Riemann zeta
function ζ(s), and gave a good and reasonable explanation that their q-analogue
maybeabestchoice. Theyalsocommentedthatq-analogueofζ(s)canbeachieved
by modifying their method. Furthermore it is clear that q-Genocchi polynomials
of weight (α,β) are in a class of orthogonal polynomials and we know that most
such special functions that are orthogonal are satisfied in multiplication theorem,
sointhispresentpaperweshowthispropertyistrueforq-Genocchipolynomialsof
weight(α,β). Inthisintroductorysection,wepresentthedefinitionsandnotations
(and some of the Important properties and characteristics) of the various special
functions, polynomials and numbers, which are potentially useful in the remainder
of the paper.
Assume that p be a fixed odd prime number. Throughoutthis paper we use the
followingnotations. ByZ wedenotetheringofp-adicrationalintegers,Qdenotes
p
the field of rational numbers, Q denotes the field of p-adic rational numbers, and
p
C denotes the completion of algebraic closure of Q . Let N be the set of natural
p p
numbers and Z =N∪{0}. Let v be the normalized exponential valuation of C
+ p p
with|p| =p−vp(p) =p−1.Whenonespeaksofq-extension,q isconsideredinmany
p
wayssuchasanindeterminate,acomplexnumberq ∈Corp-adicnumberq ∈C .If
p
q ∈Conenormallyassumethat|q|<1.Ifq ∈Cp,weassumethat|1−q|p <p−p−11
so that qx =exp(xlogq) for |x| ≤1. We use the following notation as follows:
p
1−qx 1−(−q)x
[x] = , [x] =
q 1−q −q 1+q
Note that limq→1[x]q =x; cf. [1-24].
For a fixed positive integer d with (d,f)=1, we set
X = X =limZ/dpNZ,
d ←−
N
X∗ = ∪ a+dpZ
p
0<a<dp
(a,p)=1
and
a+dpNZ = x∈X |x≡a moddpN ,
p
where a∈Z satisfies the condit(cid:8)ion 0≤a<dpN(cid:0). (cid:1)(cid:9)
ByuseKoblitz[N.Koblitz, p-adicNumbersp-adicAnalysisandZetaFunctions,
Springer-Verlag, New York Inc, 1977] notations, A p-adic distribution µ on X is a
Q -linearvectorspacehomomorphismfromtheQ -vectorspaceoflocallyconstant
p p
functions on X to Q . If f : X →Q is locally constant, instead of writing µ(f)
p p
for the value of µ at f, we usually write fµ. Also it is known that we can write
µ as follows:
q R
qx
µ x+pNZ =
q p [pN]
q
(cid:0) (cid:1)
is a distribution on X for q ∈C with |1−q| ≤1. For
p p
f ∈UD(Z )={f |f :Z →C is uniformly differentiable function},
p p p
ON THE MODIFIED q-GENOCCHI NUMBERS AND POLYNOMIALS WITH WEIGHT (α,β)3
the fermionic p-adic q-integral on Z is defined by T. Kim as follows:
p
pN−1
(1.1) I−q(f) = Z f(x)dµ−q(x)=Nl→im∞ f(x)µ−q x+pNZp
Z p x=0
X (cid:0) (cid:1)
pN−1
1
= lim (−1)xf(x)qx
N→∞[pN]−q x=0
X
Let q →1, then we have fermionic integration on Z as follows:
p
pN−1
I−1(f)= Z f(x)dµ−1(x)=Nl→im∞ (−1)xf(x),
Z p x=0
X
So by applying f(x)=ext, we get
∞
2t tn
(1.2) t etxdµ (x)= = G
Z −1 et+1 nn!
Z p n=0
X
Where G are Genocchi numbers. By using (1.2), we have
n
∞
G tn
Z extdµ−1(x)= nn++11n!
Z p n=0
X
so from above, we obtain
∞ ∞
tn G tn
xndµ (x) = n+1
n=0 ZZp −1 !n! n=0(cid:18)n+1(cid:19)n!
X X
By comparing coefficients of tn on both sides of the above equation it is fairly
n!
straightforwardto deduce,
G
nn++11 = Z xndµ−1(x).
Z p
The definition of modified q-Euler numbers are given by
[2] [2] , k =0
(1.3) ε = q, (qε+1)k−ε = q
0,q k,q
2 0, k >0
(cid:26)
with usual the convention about replacing εk by ε cf. [15],[16]. It was known
k,q
thatthe modified q-euler numbers canbe representedby p-adic q-integralonZ as
p
follows:
εn,q = q−t[t]nq dµ−q(t).
Z
Z p
In [3,14,15,17], q-Genocchi numbers are defined as follows:
[2] ,n=1
(1.4) G =0, and q(qG +1)n+G = q
0,q q n,q
0, n>1
(cid:26)
n
with the usual convention of replacing (G ) by G .
q n,q
In [6], (h,q)-Genocchi numbers are indicated as:
n [2] , n=1
G(h) =0, and qh−2 qG(h)+1 +G(h) = q
0,q q n,q 0, n>1,
(cid:16) (cid:17) (cid:26)
4 SERKANARACI,MEHMETAC¸IKGO¨Z,FENGQI,ANDHASSANJOLANY
n
with the usual convention about replacing G(h) by G(h).
q n,q
Recently, for n ∈Z+, Araci et al. are consi(cid:16)dered(cid:17)weighted q-Genocchi numbers
by
n [2] , n=1
(1.5) G(α) =0, q1−α qG(α)+1 +G(α) = q
0,q q n,q 0, n6=1,
(cid:16) (cid:17) (cid:26)
n
e e e
withtheusualconventionaboutreplacing G byG (formoreinformation,
q n,q
see [1]) (cid:16) (cid:17)
For α,n ∈ Z and h ∈ N, Araci et al. [2e] definedeweighted (h,q)-Genocchi
+
numbers as follows:
G(α,h) = q(h−1)x[x]n dµ (x).
n+1,q qα −q
Z
Z p
e
Taekyun Kim, by using p-adic q-integral on Z , introduced a new class of num-
p
bers andpolynomials. He added a weightonq-Bernoullinumbers andpolynomials
anddefined q-Bernoullinumbers with weightα. He is givensome interestingprop-
erties concerning q-Bernoulli numbers and polynomials with weight α. After, by
using p-adic q-integral on Z , severalmathematicians started to study on this new
p
branchofgeneratingfunctiontheoryandextendedmostofthesymmetricproperties
of q-Bernoulli numbers and polynomials to q-Bernoulli numbers and polynomials
with weight α (for more informations, see [1],[2],[4],[5],[6],[7],[8],[9],[10],[11],[12]).
With the same motivation, we also introduce modified q-Genocchi numbers and
polynomialswithweight(α,β). Also,we givesomeinteresting propertiesthis type
ofpolynomials. Furthermore,wederivetheq-extensionsofzetatypefunctionswith
weight (α,β) from the Mellin transformation to this generating function which in-
terpolates the q-Genocchi polynomials with weight (α,β) at negative integers.
2. Modified q-Genocchi numbers and polynomials with weight
(α,β)
In this section, we derive some interesting properties Modified q-Genocchi num-
bers and polynomials with weight (α,β).
Lemma 1. For n∈Z ,we obtain
+
n−1
(2.1) I(β) q−βxf +(−1)n−1I(β) q−βxf =[2] (−1)n−l−1f(l),
−q n −q qβ
l=0
(cid:0) (cid:1) (cid:0) (cid:1) X
Proof. Let be fn(x) = f(x+n) and I−(βq)(f) = Z f(x)dµ−qβ(x) by the (1.1),
p
we easily get
R
pN−1
1
−I−(βq) q−βxf1 = Nl→im∞[pN]−qβ x=0 f(x+1)(−1)x
(cid:0) (cid:1) X
1 pN−1 f pN +f(0)
= lim f(x)(−1)x−[2] lim
N→∞[pN]−qβ x=0 qβ N→∞ (cid:0)1+(cid:1)qβpN
X
(2.2) = I−(βq) q−βxf −[2]qβ f(0)
(cid:0) (cid:1)
ON THE MODIFIED q-GENOCCHI NUMBERS AND POLYNOMIALS WITH WEIGHT (α,β)5
and
pN−1
1
I−(βq) q−βxf2 = ZZpq−βxf(x+2)dµ−qβ(x)=Nl→im∞[pN]−qβ x=0 f(x+2)(−1)x
(cid:0) (cid:1) X
−f(0)+f(1)−f pN +f pN +1
= I−(βq) q−βxf +[2]qβNl→im∞ 1+qβpN
(cid:0) (cid:1) (cid:0) (cid:1)
= I(β)(cid:0)q−βxf(cid:1)+[2] (f(1)−f(0))
−q qβ
Thus, we have (cid:0) (cid:1)
1
I(β) q−βxf −I(β) q−βxf =[2] (−1)1−lf(l)
−q 2 −q qβ
l=0
(cid:0) (cid:1) (cid:0) (cid:1) X
By continuing this process, we arrive at the desired result. (cid:3)
Definition 1. Let α,n,β ∈ Z . We define modified q-Genocchi numbers with
+
weight (α,β) as follows:
g(α,β) ∞
n+1,q m n
(2.3) =[2] (−1) [m]
n+1 qβ qα
m=0
X
Theorem 1. For α,n,β ∈Z , we get
+
(2.4) gn(α+,β1,)q = [2]qβ n n (−1)l 1
n+1 (1−qα)n l 1+qαl
l=0(cid:18) (cid:19)
X
Proof. By (2.3), we develop as follows:
g(α,β)
n+1,q
n+1
∞
[2]
= qβ (−1)m(1−qmα)n
(1−qα)n
m=0
X
= [2]qβ ∞ (−1)m n n (−1)l(qmα)l
(1−qα)n l
m=0 l=0(cid:18) (cid:19)
X X
= [2]qβ n n (−1)l ∞ (−1)mqmαl
(1−qα)n l
l=0(cid:18) (cid:19) m=0
X X
[2]qβ n n l 1
= (−1) .
(1−qα)n l 1+qαl
l=0(cid:18) (cid:19)
X
Thus, we complete the proof of Theorem. (cid:3)
By the following Theorem,we get Witt’s type formula of this type polynomials.
Theorem 2. For β,α,n∈Z , we get
+
g(α,β)
(2.5) nn++1,1q = Z q−βx[x]nqαdµ−qβ (x).
Z p
6 SERKANARACI,MEHMETAC¸IKGO¨Z,FENGQI,ANDHASSANJOLANY
Proof. By using p-adic q-integral on Z , namely, replace f(x) by q−βx[x]n and
p qα
µ−q x+pNZp by µ−qβ x+pNZp into (1.1), we get
(cid:0) (cid:1) (cid:0) 1(cid:1) n n
Z q−βx[x]nqαdµ−qβ(x) = (1−qα)n l (−1)l Z qαlx−βxdµ−qβ(x)
Z p l=0(cid:18) (cid:19) Z p
X
n pN−1
= 1 n (−1)l lim 1 −qαl x
(1−qα)n l=0(cid:18)l(cid:19) N→∞[pN]−qβ x=0
X X (cid:0) (cid:1)
1 n n l [2]qβ 1+ qαl pN
(2.6) = (−1) lim
(1−qα)n l 1+qαl N→∞ 1+qβpN
l=0(cid:18) (cid:19) (cid:0) (cid:1)
X
= [2]qβ n n (−1)l 1
(1−qα)n l 1+qαl
l=0(cid:18) (cid:19)
X
Use of (2.4) and (2.6), we arrive at the desired result. (cid:3)
The Witt’s type formula of modified q-Genocchi numbers with weight (α,β)
asserted by Theorem 2, do aid in translating the various properties and results
involving q-Genocchi numbers with weight (α,β) which we state some of them in
g(1,1)
this section. We put α→1 and β →1 into (2.5), we readily see n+1,q =ε .
n+1 n,q
Corollary 1. Let C(α,β)(t)= ∞ g(α,β)tn. Then we have
q n=0 n,q n!
∞
P
Cq(α,β)(t)=[2]qβt (−1)met[m]qα.
m=0
X
Proof. From (2.3) we easily get,
∞
(2.7) q−βxet[x]qαdµ−qβ (x)=[2]qβ t (−1)met[m]qα
Z
Z p m=0
X
By expression (2.7), we have
∞ ∞
tn
gn(α,q,β)n! =[2]qβt (−1)met[m]qα
n=0 m=0
X X
Thus, we complete the proof of Theorem. (cid:3)
Now,weconsiderthe modifiedq-Genocchipolynomialspolynomialswithweight
α as follows:
(α,β)
g (x)
(2.8) n+1,q = q−βt[x+t]n dµ (t), n∈N and α∈Z
n+1 Z qα −qβ +
Z p
From expression (2.8), we see readily
gn(α+,1β,)q(x) = [2]qβ n n (−1)lqαlx 1
n+1 (1−qα)n l 1+qαl
l=0(cid:18) (cid:19)
X
∞
m n
(2.9) = [2] (−1) [m+x]
qβ qα
m=0
X
ON THE MODIFIED q-GENOCCHI NUMBERS AND POLYNOMIALS WITH WEIGHT (α,β)7
Let C(α,β)(t,x)= ∞ g(α,β)(x)tn. Then we have
q n=0 n,q n!
∞
P
C(α,β)(t,x) = [2] t (−1)met[m+x]qα
q qβ
m=0
X
∞
tn
(2.10) = g(α,β)(x) .
n,q n!
n=0
X
By Lemma 1, we get the following Theorem:
Theorem 3. For m∈N,and α,β,n∈Z , we get
+
g(α,β) g(α,β) (n) n−1
m+1,q +(−1)n−1 m+1,q =[2] (−1)n−l−1[l]m
m+1 m+1 qβ qα
l=0
X
Proof. ByapplyingLemma1themethodologyandtechniquesusedaboveingetting
someidentitiesforthegeneratingfunctionsofthemodifiedq-Genocchinumbersand
polynomials with weight (α,β), we arrive at the desired result. (cid:3)
Theorem 4. The following identity holds:
[2] ,if n=1,
g(α,β) =0, and g(α,β)(1)+g(α,β) = qβ
0,q n,q n,q 0, if n>1.
(cid:26)
Proof. In (2.2) it is known that
I(β) q−βxf +I(β) q−βxf =[2] f(0)
−q 1 −q qβ
If we take f(x)=et[x]qα,(cid:0)then we(cid:1)have (cid:0) (cid:1)
[2]qβ = q−βxet[x+1]qαdµ−qβ (x)+ q−βxet[x]qαdµ−q−β(x)
Z Z
Z p Z p
∞ tn−1
(2.11) = g(α,β)(1)+g(α,β)
n,q n,q n!
nX=0(cid:16) (cid:17)
Therefore, we get the Proof of Theorem. (cid:3)
Theorem 5. For d≡1(mod2), α,β ∈Z and n∈N, we get,
+
g(α,β)(dx)= [d]qnα−1 d−1(−1)ag(α,β) x+ a .
n,q [d] n,qd d
−qβ Xa=0 (cid:16) (cid:17)
Proof. From (2.8), we can easily derive the following (2.12)
ZZpq−βt[x+t]nqαdµ−qβ(t) = [d[d]−]nqqαβ da−=10(−1)aZZpq−βt(cid:20)x+d a +t(cid:21)nqdαdµ(−qd)β(t)
X
[d]n d−1 g(α,β) x+a
(2.12) = qα (−1)a n+1,qd d .
[d] n+1
−qβ a=0 (cid:0) (cid:1)
X
So,byapplyingexpression(2.12),wegetatthedesiredresultandproofiscomplete.
(cid:3)
8 SERKANARACI,MEHMETAC¸IKGO¨Z,FENGQI,ANDHASSANJOLANY
3. Interpolation function of the polynomials g(α,β)(x)
n,q
In this section, we derive the interpolation function of the generating functions
of modified q-Genocchi polynomials with weight α and we give the value of q-
extensionzetafunctionwithweight(α,β)atnegativeintegersexplicitly. Fors∈C,
by applying the Mellin transformation to (2.10), we obtain
∞
1
ξ(α,β)(s,x|q) = ts−2 −C(α,β)(−t,x) dt
Γ(s) q
Z∞0 n ∞ o
= [2]qβ (−1)m Γ1(s) ts−1e−t[m+x]qαdt
m=0 Z0
X
where Γ(s) is Euler gamma function. We have
∞ m
(−1)
ξ(α,β)(s,x|q)=[2]
qβ [m+x]s
m=0 qα
X
So, we define q-extension zeta function with weight (α,β) as follows:
Definition 2. For s∈C and α,β ∈N, we have
∞ m
(−1)
(3.1) ξ(α,β)(s,x|q)=[2]
qβ [m+x]s
m=0 qα
X
ξ(α,β)(s,x|q) can be continued analytically to an entire function.
Observe that, if q → 1, then ξ(α,β)(s,x|1) = ζ(s,x) which is the Hurwitz-
Euler zeta functions. Relation between ξ(α,β)(s,x|q) and g(α,β)(x) are given by
n,q
the following theorem:
Theorem 6. For α,β ∈N and n∈N, we get
g(α,β) (x)
ξ(α,β)(−n,x|q)= n+1,q .
n+1
Proof. By substituting s=−n into (3.1), we arrive at the desired result. (cid:3)
References
[1] Araci,S.,Erdal,D.,andSeo,J-J.,Astudyonthefermionicp-adicq-integralrepresentationon
Zp associatedwithweightedq-Bernsteinandq-Genocchipolynomials,AbstractandApplied
Analysis,Volume2011,ArticleID649248, 10pages.
[2] Araci,S., Seo, J-J.,and Erdal,D.,New construction weighted (h,q)-Genocchi numbers and
polynomialsrelatedtozetatypefunctions,DiscreteDynamicsinNatureandSociety,Volume
2011,ArticleID487490, 7pages,doi:10.1155/2011/487490.
[3] Araci, S., Erdal, D.,and Kang, D-J., Some new properties on the q-Genocchi numbers and
polynomialsassociatedwithq-Bernsteinpolynomials,HonamMathematicalJ.33(2011),no.
2,pp.261-270.
[4] Araci,S.,Aslan,N.,andSeo,J-J.,ANoteontheweightedTwistedDirichlet’stypeq-Euler
numbersandpolynomials,HonamMathematicalJ.33(2011), no.3,pp.311-320.
[5] Araci, S., A¸cıkg¨oz, M., and Seo, J-J., A study on the weighted q-Genocchi numbers and
polynomialswiththeirInterpolation Function,Accepted inHonamMathematical Journal.
[6] Araci, S., and A¸cıkg¨oz, M., and Park, K-H.,Some identities concerning the (h,q)-Genocchi
numbers and polynomials via the p-adic q-integral on Zp and q-Bernstein polynomi-
als,(submitted).
[7] Kim,T., Onthe weighted q-Bernoullinumbers and polynomials, Advanced Studies inCon-
temporaryMathematics 21(2011), no.2,p.207-215,http://arxiv.org/abs/1011.5305.
ON THE MODIFIED q-GENOCCHI NUMBERS AND POLYNOMIALS WITH WEIGHT (α,β)9
[8] Kim, T., Lee, S. H., Dolgy, D. V., Ryoo, C. S., A note on the generalized q-Bernoulli
measures with weight α, Abstract and Applied Analysis, Article ID 867217, 9 pages,
doi:10.1155/2011/867217.
[9] Kim,T.,Choi, J., Kim,Y.H., Ryoo, C. S., Anote onthe weighted p-adicq-Eulermeasure
onZp,Advn.Stud.Contemp. Math.21(2011), 35-40.
[10] Hwang,K-W.,Dolgy, D-V.,Lee, S.H.,andKim,T.,OntheHigher-Orderq-Eulernumbers
and polynomials with weight α, Discrete Dynamics in Nature and Society, Volume (2011),
ArticleID354329, 12pages.
[11] Kim,T.,andChoi,J.,Ontheq-Bernoullinumbersandpolynomialswithweightα,Abstract
andAppliedAnalysis,Volume2011(2011), ArticleID392025, 14pages.
[12] Kim, T., Dolgy, D-V., Lee, B., and Rim S-H., Identities on the weighted q-Euler numbers
of higher order, Discrete Dynamics inNature and Society, Volume 2011 (2011), ArticleID
918364,6pages.
[13] Kim,T.,Ontheq-extensionofEulerandGenocchinumbers,J.Math.Anal.Appl.326(2007)
1458-1465.
[14] Kim, T., On the multiple q-Genocchi and Euler numbers, Russian J. Math. Phys. 15 (4)
(2008)481-486. arXiv:0801.0978v1[math.NT]
[15] Kim, T., The modified q-Euler numbers and polynomials, Advn. Stud. Contemp. Math. 16
(2008), 161-170.
[16] Ozden,H.,Simsek,Y.,Rim,S-H.,Cangul,I-N.,Anoteonp-adicq-eulermeasure,Adv.Stud.
Contemp.Math.14(2007), 233-239.
[17] Kim,T.,q-Volkenbornintegration,Russ.J.Math.phys.9(2002),288-299.
[18] Kim,T.,q-BernoullinumbersandpolynomialsassociatedwithGaussianbinomialcoefficients,
Russ.J.Math.Phys.15(2008),51-57.
[19] Kim, T., An invariant p-adic q-integrals on Zp, Applied Mathematics Letters, vol. 21, pp.
105-108,2008.
[20] Kim, T., A Note on the q-Genocchi Numbers and Polynomials, Journal of Inequalities and
Applications,ArticleID71452, 8pages,doi:10.1155/2007/71452.
[21] Kim,T.Choi,J.Kim,Y.H.andJang,L.C.,Onp-AdicAnalogueofq-BernsteinPolynomials
andRelatedIntegrals,DiscreteDynamicsinNatureandSociety,ArticleID179430,9pages,
doi:10.1155/2010/179430.
[22] Kim, T., q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear
Math.Phys.,14(2007), no.1,15–27.
[23] Kim, T., New approach to q-Euler polynomials of higher order, Russ. J. Math. Phys., 17
(2010), no.2,218–225.
[24] Kim,T., Someidentities onthe q-Eulerpolynomialsof higher orderand q-Stirlingnumbers
bythefermionicp-adicintegralonZp,Russ.J.Math.Phys.,16(2009), no.4,484–491.
UniversityofGaziantep,FacultyofScienceandArts,DepartmentofMathematics,
27310Gaziantep,TURKEY
E-mail address: [email protected]
UniversityofGaziantep,FacultyofScienceandArts,DepartmentofMathematics,
27310Gaziantep,TURKEY
E-mail address: [email protected]
Department of Mathematics, College of Science, Tianjin Polytechnic University,
Tianjin300160,China
E-mail address: [email protected]
SchoolofMathematics,StatisticsandComputerScience,UniversityofTehran,Iran
E-mail address: [email protected]