A unification of the multiple twisted Euler and Genocchi numbers and polynomials associated with p adic q integral on Zp at q=-1

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by  Serkan Araci

A UNIFICATION OF THE MULTIPLE TWISTED EULER AND
GENOCCHI NUMBERS AND POLYNOMIALS ASSOCIATED
WITH p-ADIC q-INTEGRAL ON Z AT q =−1
p
2
1
0 SERKANARACI,MEHMETACIKGOZ,KYOUNG-HOPARK,ANDHASSANJOLANY
2

n Abstract. The present paper deals with unification of the multiple twisted
a EulerandGenocchinumbersandpolynomialsassociatedwithp-adicq-integral
J on Zp at q =−1. Some earlier results of Ozden’s papers in terms of unifica-
5 tion of the multiple twisted Euler and Genocchi numbers and polynomials
associated withp-adicq-integral onZp at q=−1can bededuced. We apply
] themethodofgeneratingfunctionandp-adicq-integralrepresentationonZp,
T whichareexploitedtoderivefurtherclassesofEulerpolynomialsandGenoc-
N chi polynomials. To be moreprecise we summarizeour results as follows,we
obtain some relations between H.Ozden’s generating function and fermionic
h. p-adicq-integralonZp atq=−1. FurthermorewederiveWitt’stypeformula
t for the unification of twisted Euler and Genocchi polynomials. Also we de-
a
rivedistributionformula(MultiplicationTheorem)formultipletwistedEuler
m
and Genocchi numbers and polynomials associated with p-adic q-integral on
[ Zp at q =−1which yields a deeper insightinto the effectiveness of this type
of generalizations. Furthermorewedefine unification ofmultipletwistedzeta
1 function and we obtain an interpolation formula between unification of mul-
v tiple twisted zeta function and unification of the multiple twisted Euler and
5 Genocchinumbersatnegativeinteger. Ournewgeneratingfunctionpossessa
1 numberofinterestingpropertieswhichwestateinthispaper.
3
1
.
1
0 1. Introduction, Definitions and Notations
2
1 Bernoulli numbers introduced by Jacques Bernoulli (1654-1705), in the second
:
v partof his treatise published in 1713,Ars conjectandi, at the time, Bernoulli num-
i bers were used for writing the infinite series expansions of hyperbolic and trigono-
X
metric functions. Van den berg was the firstto discuss finding recurrence formulae
r
a for the Bernoulli numbers with arbitrary sized gaps (1881). Ramanujan showed
how gaps of size 7 could be found, and explicitly wrote out the recursion for gaps,
of size 6. Lehmer in 1934 extended these methods to Euler numbers, Genocchi
numbers,andLucasnumbers(1934),andcalculatedthe 196-thBernoullinumbers.
The study of generalized Bernoulli, Euler and Genocchi numbers and polynomials
and their combinatorial relations has received much attention [20], [21], [22]-[25],
[26],[27],[28],[29],[30]. GeneralizedBernoullipolynomials,generalizedEulerpoly-
nomials and generalized Genocchi numbers and polynomials are the signs of very
strong bond between elementary number theory, complex analytic number theory,
Homotopy theory (stable Homotopy groups of spheres), differential topology (dif-
ferentialstructures onspheres),theory ofmodular forms (Eisensteinseries), p-adic
Date:October 26,2011.
1991 Mathematics Subject Classification. 05A10,11B65,28B99,11B68,11B73.
Key words and phrases.
1
2 SERKANARACI,MEHMETACIKGOZ,KYOUNG-HOPARK,ANDHASSANJOLANY
analytic number theory (p-adic L-functions), quantum physics(quantum Groups).
p-adic numbers were invented by Kurt Hensel around the end of the nineteenth
century. In spite of their being already one hundred years old, these numbers are
still today enveloped in an aura of mystery within the scientific community. The
p-adicintegralwasusedinmathematicalphysics,forinstance,thefunctionalequa-
tion of the q-zeta function, q-stirling numbers and q-Mahler theory of integration
with respect to the ring Z together with Iwasawa’sp-adic q-L functions. Also the
p
p-adic interpolation functions of the Bernoulli and Euler polynomials have been
treated by Tsumura [31] and Young [32]. Professor T.Kim [3]-[17] also studied on
p-adic interpolation functions of these numbers and polynomials. In [33], Carlitz
originally constructed q-Bernoulli numbers and polynomials. These numbers and
polynomials are studied by many authors (see cf. [3]-[19], [34], [35], [38]). In the
lastdecade,a surprisingnumberofpapers appearedproposingnew generalizations
of the Bernoulli, Euler and Genocchi polynomials to real and complex variables.
In [3]-[18], Kim studied some families of multiple Bernoulli, Euler and Genocchi
numbers and polynomials. By using the fermionic p-adic invariant integral on Z ,
p
he constructed p-adic Bernoulli, Euler and Genocchi numbers and polynomials of
higher order. A unification (and generalization) of Bernoulli polynomials and Eu-
ler polynomials with a,b and c parameters first was introduced and investigated
by Q.-M.Luo [22], [23], [24], [25]. After he with H.M.Srivastava defined unifica-
tion (and generalization) of Apostol type Bernoulli polynomials with a, b and c
parametersofhigherorder[25]. AfterHacerOzdenetal[35]. unifiedandextended
the generating functions of the generalized Bernoulli polynomials, the generalized
Euler polynomials and the generalized Genocchi polynomials associated with the
positive real parameters a and b and the complex parameter. Also they by ap-
plying the Mellin transformation to the generating function of the unification of
Bernoulli, Euler and Genocchi polynomials, constructed a unification of the zeta
functions. Actually their definition provides a generalization and unification of
the Bernoulli, Euler and Genocchi polynomials and also of the Apostol–Bernoulli,
Apostol–EulerandApostol–Genocchipolynomials,whichwereconsideredinmany
earlier investigations by (among others) Srivastava et al. [39], [40], [41], Karande
[42]. Also they by using a Dirichlet character defined unification of the generating
functions of the generalized Bernoulli, Euler and Genocchi polynomials and num-
bers. T.Kimin[13],constructedApostol-Eulernumbersandpolynomialsbyusing
fermionicexpressionofp-adicq-integralatq =−1. Inthispaperbyhismethodwe
deriveseveralpropertiesfor unificationofthe multiple twistedEuler andGenocchi
numbers and polynomials.
Letp be afixed oddprime number. Throughoutthis paperwe use the following
notations, by Z denotes the ring of p-adic rational integers, Q denotes the field of
p
rational numbers, Q denotes the field of p-adic rational numbers, and C denotes
p p
the completionofalgebraicclosureof Q . LetN be the setof naturalnumbers and
p
N∗ = N∪{0}. The p-adic absolute value is defined by |p| = 1. In this paper we
p p
assume |q−1| <1 as an indeterminate. [x] is a q-extensionof x which is defined
p q
by [x] = 1−qx, we note that lim [x] =x.
q 1−q q→1 q
We say that f is a uniformly differntiable function at a point a ∈ Z , if the
p
difference quotient
f(x)−f(y)
F (x,y)=
f
x−y
A UNIFICATION OF MULTIPLE TWISTED EULER AND GENOCCHI POLYNOMIALS 3
has a limit f´(a) as (x,y)→(a,a) and denote this by f ∈UD(Z ).
p
Let UD(Z ) be the set of uniformly differentiable function on Z . For f ∈
p p
UD(Z ), let us begin with the expressions
p
1
f(x)qx = f(x)µ x+pNZ ,
[pN] q p
0≤Xx<pN 0≤Xx<pN (cid:0) (cid:1)
represents p-adic q-analogue of Riemann sums for f. The integral of f on Z
p
will be defined as the limit (N →∞) of these sums, when it exists. The p-adic
q-integral of function f ∈UD(Z ) is defined by Kim
p
pN−1
1
(1.1) I (f)= f(x)dµ (x)= lim f(x)qx
q ZZp q N→∞[pN]q x=0
X
The bosonic integral is considered by Kim as the bosonic limit q →1, I (f)=
1
lim I (f). Similarly, the fermionic p-adic integral on Z is considered by Kim
q→1 q p
as follows:
I (f)= lim I (f)= f(x)dµ (x)
−q q −q
q→−q Z
Z p
Assume that q → 1, then we have fermionic p-adic fermionic integral on Z as
p
follows
pN−1
x
(1.2) I (f)= lim I (f)= lim f(x)(−1) .
−1 q
q→−1 N→∞
x=0
X
If we take f (x)=f(x+1) in (1.2), then we have
1
(1.3) I (f )+I (f)=2f(0).
−1 1 −1
Let p be a fixed prime. For a fixed positive integer d with (p,d)=1, we set
X = X =limZ/dpNZ,
d
←
N
X = Z ,
1 p
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)
Definition 1. (see, cf. [36]). A unification y (x:k,a,b) of the Bernoulli, Euler
n,β
and Genochhi polynomials is given by the following generating function:
2 t k ∞ tn β
F (x;t;k,β) = 2 ext = y (x:k,a,b) t+log <2π; x∈R
a,b βbet−ab n,β n! a
(cid:0) (cid:1) n=0 (cid:18)(cid:12) (cid:18) (cid:19)(cid:12) (cid:19)
X (cid:12) (cid:12)
(1.4) k ∈N∗;a,b∈R+;β ∈C , (cid:12) (cid:12)
(cid:12) (cid:12)
where as usual R+,(cid:0)and C denote the sets o(cid:1)f positive real numbers and complex
numbers, respectively, R being the set of real numbers.
4 SERKANARACI,MEHMETACIKGOZ,KYOUNG-HOPARK,ANDHASSANJOLANY
Observe that, if we put x = 0 in the generating function (1.4), then we obtain
the corresponding unification of the generating functions of Bernoulli, Euler and
Genocchi numbers. Then we have
y (0:k,a,b)=y (k,a,b).
n,β n,β
We are now ready to give relationship between the Ozden’s generating function
and the fermionic p-adic q-integral on Z at q =−1 with the following theorem:
p
Theorem 1. The following relationship holds:
t k β bx ∞ tn
(1.5) a−b (−1)x+1 etxdµ (x)= y (k,a,b) .
2 Z a −1 n,β n!
(cid:18) (cid:19) Z p (cid:18) (cid:19) n=0
X
Proof. We set f(x)=a−b t k(−1)x+1 β bxetx in (1.3), it is easy to show
2 a
t k β b (cid:0) (cid:1) (cid:16) (cid:17) β bx 2 t k
a−b − et+1 (−1)x+1 etxdµ (x) = − 2
(cid:18)2(cid:19) (cid:18)a(cid:19) !ZZp (cid:18)a(cid:19) −1 (cid:0)ab(cid:1)
t k β bx 2 t k
a−b (−1)x+1 etxdµ (x) = 2
(cid:18)2(cid:19) ZZp (cid:18)a(cid:19) −1 βbe(cid:0)t−(cid:1)ab
So, we complete the proof. (cid:3)
Theorem 2. Then the following identity holds:
bx
β (n−k)!
(−1)x+1 xn−kdµ (x)=2kab y (k,a,b).
Z a −1 n! n,β
Z p (cid:18) (cid:19)
Proof. From (1.5) and by using the taylor expansion of etx, we readily see that,
∞ β bx tn+k ∞ tn
2−ka−b (−1)x+1 xndµ (x) = y (k,a,b)
n=0 ZZp (cid:18)a(cid:19) −1 ! n! n=0 n,β n!
X X
Bycomparingcoefficientsoftn inthebothsidesoftheaboveequation,wearrive
at the desired result. (cid:3)
Similarly, we obtain te following theorem for a unification of the Euler and
Genocchi polynomials as follows:
Theorem 3. Then the following identity holds:
by
β n!
(1.6) (−1)y+1 (x+y)ndµ (y)=2kab y (x:k,a,b).
Z a −1 (n+k)! n+k,β
Z p (cid:18) (cid:19)
From the binomial theorem in (1.6), we possess the following theorem:
Theorem 4. The following relation holds:
n n
y (x:k,a,b)
n+k,β = m y (k,a,b)xn−m
n+k m+k m+k,β
k m=0 (cid:0)k(cid:1)
X
Proof. By using (1.6)(cid:0)and(cid:1)binomial theor(cid:0)em, (cid:1)we express the following relation
n by
n β n!
(−1)y+1 ymdµ (y) xn−m =2kab y (x:k,a,b)
m=0(cid:18)m(cid:19) ZZp (cid:18)a(cid:19) −1 ! (n+k)! n+k,β
X
A UNIFICATION OF MULTIPLE TWISTED EULER AND GENOCCHI POLYNOMIALS 5
By using p-adicq-integralon Z at q =−1, we arriveat the desiredproofofthe
p
theorem. (cid:3)
Now, we consider symmetric properties of this type of polynomials as follows:
Theorem 5. The following relation holds:
y 1−x:k,a−1,b =(−1)k+n+1βbaby (x:k,a,b).
n,β−1 n,β
Proof. We set x→(cid:0)1−x, β →β−1(cid:1)and a→a−1 into (1.6), that is
β−1 by
(−1)y+1 (1−x+y)ndµ (y)
Z a−1 −1
Z p (cid:18) (cid:19)
−by
β
= (−1)n (−1)y+1 (x−1+y)ndµ (y)
Z a −1
Z p (cid:18) (cid:19)
= (−1)k+n+1βbaby (x:k,a,b)
n,β
Thus, we complete proof of the theorem. (cid:3)
Ozdenhasobtaineddistributionformulafory (x:k,a,b).Wewillalsoobtain
n,β
distribution formula by using p-adic q-integral on Z at q =−1.
p
Theorem 6. The following identity holds:
d−1 bj
β x+j
y (x:k,a,b)=ab(d−1)dn−k y :k,ad,b .
n,β a n,βd d
j=0(cid:18) (cid:19) (cid:18) (cid:19)
X
Proof. By using definition of the p-adic integral on Z , we compute
p
by
n! β
2kab y (x:k,a,b) = (−1)y+1 (x+y)ndµ (y)
(n+k)! n+k,β Z a −1
Z p (cid:18) (cid:19)
dpN−1 by
β
= lim (−1)y+1 (x+y)n(−1)y
N→∞ a
y=0 (cid:18) (cid:19)
X
d−1 bj pN−1 bdy n
β β x+j
= dn lim (−1)y+1 +y (−1)y
a N→∞ a d
j=0(cid:18) (cid:19) y=0 (cid:18) (cid:19) (cid:18) (cid:19)
X X
d−1 β bj βd by x+j n
= dn (−1)y+1 +y dµ (y)
j=0(cid:18)a(cid:19) ZZp ad! (cid:18) d (cid:19) −1
X
d−1 bj
β n! x+j
= dn 2kadb y :k,ad,b
a (n+k)! n+k,βd d
j=0(cid:18) (cid:19) (cid:18) (cid:19)
X
Substituting n by n−k, we will be completed the proof of theorem. (cid:3)
Remark1. Thisdistributionfory (x:k,a,b)isalsointroducedbyOzdencf.[36].
n,β
Definition 2. (see, for detail [35])Let χ be a Dirichlet character with conductor
d ∈ N. The generating functions of the generalized Bernoulli, Euler and Genocchi
6 SERKANARACI,MEHMETACIKGOZ,KYOUNG-HOPARK,ANDHASSANJOLANY
polynomials with parameters a, b, β and k have been defined by Ozden, Simsek and
Srivastava as follows:
(1.7)F (t,k,a,b)
χ,β
j
k d χ(j) β ejt
t a
= 2
2 βbde(cid:16)dt−(cid:17)abd
(cid:18) (cid:19) j=1
X
∞ tn β
= y (x:k,a,b) , t+blog <2π; d,k ∈N; a,b∈R+; β ∈C
n,χ,β
n! a
n=0 (cid:18)(cid:12) (cid:18) (cid:19)(cid:12) (cid:19)
X (cid:12) (cid:12)
By using p-adic integral on Z ,(cid:12)we can obtain(cid:12)(1.7) with the following theorem:
p(cid:12) (cid:12)
Theorem 7. Let χ be a Dirichlet’s character with conductor d ∈ N. Then the
following relation holds
(1.8)
bj
k bx d χ(j) β etj
ab(1−d) t χ(x)(−1)x+1 β etxdµ (x)=21−ktk a
2 Z a −1 βdbe(cid:16)dt−(cid:17)adb
(cid:18) (cid:19) Z p (cid:18) (cid:19) j=1
X
Proof. From the definition of p-adic q-integral on Z at q =−1, we compute
p
k bx
t β
ab(1−d) χ(x)(−1)x+1 etxdµ (x)
2 Z a −1
(cid:18) (cid:19) Z p (cid:18) (cid:19)
k dpN−1 bx
t β
= ab(1−d) lim χ(x)(−1)x+1 etx(−1)x
2 N→∞ a
(cid:18) (cid:19) x=0 (cid:18) (cid:19)
X
1 d β bj 1 td k pN−1 βd bx
= χ(j) etj lim (−1)x+1 etdx(−1)x
dk j=1 (cid:18)a(cid:19) adb (cid:18) 2 (cid:19) N→∞ x=0 ad! 
X X
1 d β bj  2 td k 
= χ(j) etj 2
dk j=1 (cid:18)a(cid:19) βdbe(cid:0)dt−(cid:1)adb!
X
bj
d χ(j) β etj
= 21−ktk a
βdbe(cid:16)dt−(cid:17)adb
j=1
X
Thus, we arrive at the desired result. (cid:3)
By expression of (1.8), we get the following equation
(1.9)
t k β bx ∞ tn
ab(1−d) χ(x)(−1)x+1 etxdµ (x)= y (x:k,a,b) .
2 Z a −1 n,χ,β n!
(cid:18) (cid:19) Z p (cid:18) (cid:19) n=0
X
WearenowreadytogivedistributionformulaforgeneralizedEulerandGenocchi
polynomials by using p-adic q-integral on Z at q =−1 by means of theorem.
p
Theorem 8. For any n,k,d∈N a,b∈R+; β ∈C, we have
d−1 bj
β x+j
y (x:k,a,b)=dn−k χ(j) y :k,ad,b .
n,χ,β a n,βd d
j=0 (cid:18) (cid:19) (cid:18) (cid:19)
X
A UNIFICATION OF MULTIPLE TWISTED EULER AND GENOCCHI POLYNOMIALS 7
Proof. By expression of (1.9), we compute as follows:
∞ tn
y (x:k,a,b)
n,χ,β
n!
n=0
X
k by
t β
= ab(1−d) χ(y)(−1)y+1 et(x+y)dµ (y)
2 Z a −1
(cid:18) (cid:19) Z p (cid:18) (cid:19)
k dpN−1 by
t β
= ab(1−d) lim χ(y)(−1)y+1 et(x+y)(−1)y
2 N→∞ a
(cid:18) (cid:19) y=0 (cid:18) (cid:19)
X
= 1 d−1χ(j) β bj 1 dt k lim pN−1(−1)y+1 βd byedt(x+dj+y)(−1)y
dk j=0 (cid:18)a(cid:19) adb (cid:18) 2 (cid:19) N→∞ y=0 ad! 
X X
= d1k dj=−01χ(j)(cid:18)βa(cid:19)bja1db (cid:18)d2t(cid:19)kZZp(−1)y+1 βadd!byedt(x+dj+y)dµ−1(y) 
X
1 d−1 β bj∞ x+j tn 
= χ(j) dny :k,ad,b
dk a n,βd d n!
j=0 (cid:18) (cid:19) n=0 (cid:18) (cid:19) !
X X
∞ d−1 β bj x+j tn
= dn−k χ(j) y :k,ad,b .
 a n,βd d n!
n=0 j=0 (cid:18) (cid:19) (cid:18) (cid:19)
X X
 
So, we complete the proof of theorem. (cid:3)
2. New properties on the unification of multiple twisted Euler and
Genocchi polynomials
In this section, we introduce a unification of the twisted Euler and Genocchi
polynomials. Weassumethatq ∈C with|1−q| <1.Forn∈N,bythedefinition
p p
of the p-adic integral on Z , we have
p
n−1
(2.1) I (f )+(−1)n−1I (f)=2 f(x)(−1)n−1−x
−1 n −1
x=0
X
where f (x)=f(x+n).
n
Let T = ∪ C = lim C = C be the locally constant space, where
p pn n→∞ pn p∞
n≥1
C = w |wpn =1 is the cylic group of order pn. For w ∈ T , we denote the
pn p
locally constant function by
(cid:8) (cid:9)
(2.2) φ :Z →C , x→wx,
w p p
If we set f(x)=φ (x)a−b t k(−1)x+1 β bxetx, then we have
w 2 a
(cid:0) (cid:1) (cid:16) (cid:17)
t k β bx 2 t k
(2.3) a−b φ (x)(−1)x+1 etxdµ (x)= 2
(cid:18)2(cid:19) ZZp w (cid:18)a(cid:19) −1 wβb(cid:0)et(cid:1)−ab
8 SERKANARACI,MEHMETACIKGOZ,KYOUNG-HOPARK,ANDHASSANJOLANY
We nowdefineunificationoftwistedEulerandGenocchipolynomialsasfollows:
2 t k ∞ tn
2 = y (k,a,b) ,
wβbet−ab n,w,β n!
(cid:0) (cid:1) n=0
X
Wenotethatbysubstitutingw=1,weobtainOzden’sgeneratingfunction(1.4).
From(2.2)and(2.3),weobtainwitt’stypeformulaforaunificatonoftwistedEuler
and Genocchi polynomials as follows:
bx
β y (k,a,b)
(2.4) a−b2−k φ (x)(−1)x+1 xndµ (x)= n+k,w,β
Z w a −1 k! n+k
Z p (cid:18) (cid:19) k
(cid:0) (cid:1)
for each w ∈T and n∈N.
p
We now establish Witt’s type formula for the unification of multiple twisted
Euler and Genocchi polynomials by the following theorem.
Definition 3. Let be w ∈T , n,h,k ∈N a,b∈R+; β ∈C, we define
p
(2.5) a−hb2−hk ... φ (x +...+x )(−1)x1+...+xh+h
w 1 h
Z Z
Z p Z p
h−times
× β b(x|1+..{.+zxh)}(x +...+x )ndµ (x )...dµ (x )
a 1 h −1 1 −1 h
(cid:18) (cid:19)
y(h) (k,a,b)
n+kh,w,β
= .
(kh)! n+kh
kh
(cid:0) (cid:1)
Remark 2. Taking h = 1 into (2.5), we get the unification of the twisted Euler
and Genocchi polynomials y (k,a,b).
n,w,β
Remark 3. By substituting h = 1 and w = 1, we obtain a special case of the
unification of Euler and Genocchi polynomials y (k,a,b).
n,β
Theorem 9. For any w ∈T , n,h,k∈N a,b∈R+; β ∈C,
p
y(h) (k,a,b) n! h y(h) (k,a,b)
n+kh,w,β = li+kh,w,β
(kh)! n+kh l1!...lh! (kh)! li+kh
kh l1+.X..+lh=n iY=1 kh
l1,...,lh≥0
(cid:0) (cid:1) (cid:0) (cid:1)
Proof. ByusingdefinitionofthemultipletwistedaunificationofEulerandGenoc-
chinumbersandpolynomials,and,definitionof(x +x +...+x )n = n! xl1xl2...xlh,
1 2 h l1+...+lh=n l1!...lh! 1 2 h
l1,...,lh≥0
P
A UNIFICATION OF MULTIPLE TWISTED EULER AND GENOCCHI POLYNOMIALS 9
we see that,
β b(x1+...+xh)
a−hb2−hk ... φ (x +...+x )(−1)x1+...+xh+h
Z Z w 1 h a
Z p Z p (cid:18) (cid:19)
h−times
×(x+x +...+x )ndµ (x )...dµ (x )
1| {z }h −1 1 −1 h
n! β bx1
= a−b2−k wx1 xl1dµ (x ) ×
l1+.X..+lh=nl1!...lh! ZZp (cid:18)a(cid:19) 1 −1 1 !
l1,...,lh≥0
β bxh
...× a−b2−kZZpwxh(cid:18)a(cid:19) xlhhdµ−1(xh)!
n! h y(h) (k,a,b)
= li+kh,w,β
l1!...lh! (kh)! li+kh
l1+.X..+lh=n jY=1 kh
l1,...,lh≥0
(cid:0) (cid:1)
Thus, we arrive at the desired result. (cid:3)
From these formulas, we can define the unification of the twisted Euler and
Genocchi polynomials as follows:
2 t k h ∞ tn
(2.6) 2 ext = y(h) (x:k,a,b) ,
wβb(cid:0)et(cid:1)−ab! n=0 n,w,β n!
X
So from above, we get the Witt’s type formula for y(h) (x:k,a,b) as follows.
n,w,β
Theorem 10. For any w∈T , n,h,k ∈N a,b∈R+; β ∈C, we get
p
β b(x1+...+xh)
a−hb2−hk ... φ (x +...+x )(−1)x1+...+xh+h
Z Z w 1 h a
Z p Z p (cid:18) (cid:19)
h−times
×(x+x +...+x )ndµ (x )...dµ (x )
1| {z }h −1 1 −1 h
(h)
y (x:k,a,b)
(2.7) = n+kh,w,β
(kh)! n+kh
kh
Note that (cid:0) (cid:1)
n!
(2.8) (x+x +x +...+x )n = xl1xl2...(x+x )lh
1 2 h l !...l ! 1 2 h
1 h
l1+.X..+lh=n
l1,...,lh≥0
We obtain the sum of powers of consecutive a unification of multiple twisted
Euler and Genocchi polynomials as follows:
Theorem 11. For any w∈T , n,h,k ∈N a,b∈R+; β ∈C, we get
p
y(h) (x:k,a,b) n! y(h) (x:k,a,b)h−1y(h) (k,a,b)
n+kh,w,β = lh+kh,w,β li+kh,w,β .
(kh)! n+kh l1!...lh! (kh)! lh+kh (kh)! li+kh
kh l1+.X..+lh=n kh jY=1 kh
l1,...,lh≥0
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
10 SERKANARACI,MEHMETACIKGOZ,KYOUNG-HOPARK,ANDHASSANJOLANY
Proof. By (2.7) and (2.8), we see that,
β b(x1+...+xh)
a−hb2−hk ... φ (x +...+x )(−1)x1+...+xh+h
Z Z w 1 h a
Z p Z p (cid:18) (cid:19)
h−times
n
×(x+x +...+x ) dµ (x )...dµ (x )
1| {z }h −1 1 −1 h
n! β bx1
= a−b2−k wx1 xl1dµ (x ) ×
l1+.X..+lh=nl1!...lh! ZZp (cid:18)a(cid:19) 1 −1 1 !
l1,...,lh≥0
β bxh
...× a−b2−kZZpwxh(cid:18)a(cid:19) (x+xh)lhdµ−1(xh)!
n! y(h) (x:k,a,b)h−1y(h) (k,a,b)
= lh+kh,w,β li+kh,w,β
l1!...lh! (kh)! lh+kh (kh)! li+kh
l1+.X..+lh=n kh jY=1 kh
l1,...,lh≥0
(cid:0) (cid:1) (cid:0) (cid:1)
So, we complete the proof of the theorem. (cid:3)
3. A unification of multiple twisted Zeta functions
Our goal in this section is to establish a unification of multiple twisted zeta
functionswhichinterpolatesofaunificationofmultipletwistedEulerandGenocchi
polynomials at negative integers. For q ∈ C, |q| < 1 and w ∈ T , a unification of
p
multiple twisted Euler and Genocchi polynomials are considered as follows:
2 t k h ∞ tn β b
(3.1) 2 = y(h) (k,a,b) , t+log w <2π.
wβb(cid:0)et(cid:1)−ab! nX=0 n,w,β n! (cid:12)(cid:12) (cid:18)a(cid:19) !(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
By (3.1), we easily see that, (cid:12) (cid:12)
∞ tn t kh 1 1
y(h) (k,a,b) = 2h ...
n,w,β n! 2 wβbet−ab wβbet−ab
n=0 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
X
t kh ∞ β bn1 ∞ β bnh
= 2h (−1)h wn1 en1t... wnh enht
2 a a
(cid:18) (cid:19) nX1=0 (cid:18) (cid:19) nXh=0 (cid:18) (cid:19)
t kh ∞ β b(n1+...+nh)
= 2h 2 (−1)h φw(n1+...+nh) a e(n1+...+nh)t
(cid:18) (cid:19) n1,.X..,nh=0 (cid:18) (cid:19)
By using the taylor expansion of e(n1+...+nh)t and by comparing the coefficients
of tn in the both side of the above equation, we obtain that
(3.2)
y(h) (k,a,b) ∞ β b(n1+...+nh)
n+kh,w,β =2h(1−k)(−1)h φ (n +...+n ) (n +...+n )n
(kh)! n+kh w 1 h a 1 h
kh n1,.X..,nh≥0 (cid:18) (cid:19)
n1+...+nh6=0
(cid:0) (cid:1)