arXiv:1001.0034v1  [math.NT]  4 Jan 2010NEW IDENTITIES INVOLVING q-EULER
POLYNOMIALS OF HIGHER ORDER
T. Kim AND Y. H. Kim
Abstract. In this paper, we present new generating functions which are relat ed to
q-Euler numbers and polynomials of higher order. From these genera ting functions, we
give new identities involving q-Euler numbers and polynomials of higher order.
§1. Introduction/ Preliminaries
LetCbe the complex number field. We assume that q∈Cwith|q|<1 and
theq-number is defined by [ x]q=1−qx
1−qin this paper. The q-factorial is given by
[n]q! = [n]q[n−1]q···[2]q[1]qand theq-binomial formulae are known that
(x:q)n=n/productdisplay
i=1(1−xqi−1) =n/summationdisplay
i=0/parenleftbiggn
i/parenrightbigg
qq(i
2)(−x)i,(see [3, 14, 15]) ,
and
1
(x:q)n=n/productdisplay
i=1/parenleftbigg1
1−xqi−1/parenrightbigg
=∞/summationdisplay
i=0/parenleftbiggn+i−1
i/parenrightbigg
qxi,(see [3, 5, 14, 15]) ,
where/parenleftbign
i/parenrightbig
q=[n]q!
[n−i]q![i]q!=[n]q[n−1]q···[n−i+1]q
[i]q!.
The Euler polynomials are defined by2
et+1ext=/summationtext∞
n=0En(x)tn
n!, for|t|< π. In the
special case x= 0,En(=En(0)) are called the n-th Euler numbers. In this paper, we
consider the q-extensions of Euler numbers and polynomials of higher orde r. Barnes’
multiple Bernoulli polynomials are also defined by
(1)
tr
/producttextr
j=1(eajt−1)ext=∞/summationdisplay
n=0Bn(x,r|a1,···,ar)tn
n!,where|t|<max
1≤i≤r2π
|ai|, (see [1, 14]).
Key words and phrases. : multiple q-zeta function, q-Euler numbers and polynomials, higher
order q-Euler numbers, Laurent series, Cauchy integral.
2000 AMS Subject Classification: 11B68, 11S80
The present Research has been conducted by the research Grant of Kw angwoon University in 2010
Typeset by AMS-TEX
1In one of an impressive series of papers (see [1, 6, 14]), Barn es developed the so-called
multiple zeta and multiple gamma function. Let a1,···,aNbe positive parameters.
Then Barnes’ multiple zeta function is defined by
ζN(s,w|a1,···,aN) =/summationdisplay
m1,···,mN=0(w+m1a1+···+mNaN)−s,(see [1]),
whereℜ(s)> N,ℜ(w)>0. Form∈Z+, we have
ζN(−m,w|a1,···,aN) =(−1)mm!
(N+m)!BN+m(w,N|a1,···,aN).
In this paper, we consider Barnes’ type multiple q-Euler numbers and polynomials.
The purpose of this paper is to present new generating functi ons which are related
toq-Euler numbers and polynomials of higher order. From the Mel lin transformation
of these generating functions, we derive the q-extensions o f Barnes’ type multiple
zeta functions, which interpolate the q-Euler polynomials of higher order at negative
integer. Finally, we give new identities involving q-Euler numbers and polynomials of
higher order.
§2.q-Euler numbers and polynomials of higher order
In this section, we assume that q∈Cwith|q|<1. Letx,a1,... ,a rbe complex
numbers with positive real parts. Barnes’ type multiple Eul er polynomialsare defined
by
(2)2r
/producttextr
j=1(eajt+1)ext=∞/summationdisplay
n=0E(r)
n(x|a1,... ,a r)tn
n!,for|t|<max
1≤i≤rπ
|wi|,(see [6]),
andE(r)
n(a1,... ,a r)(=E(r)
n(0|a1,... ,a r)) are called the n-th Barnes’ type multiple
Euler numbers. First, we consider the q-extension of Euler polynomials. The q-Euler
polynomials are defined by
(3)Fq(t,x) =∞/summationdisplay
n=0En,q(x)tn
n!= [2]q∞/summationdisplay
m=0(−q)me[m+x]qt,(see [8, 11, 13, 14, 15]) .
From (3), we have
En,q(x) =[2]q
(1−q)nn/summationdisplay
l=0/parenleftbiggn
l/parenrightbigg(−1)lqlx
(1+ql+1).
In the special case x= 0,En,q(=En,q(0)) are called the n-thq-Euler numbers. From
(3), we can easily derive the following relation.
E0,q= 1,andq(qE+1)n+En,q= 0 ifn≥1,(see [8, 16, 17]) ,
2where we use the standard convention about replacing EkbyEk,q.It is easy to show
that
lim
q→1Fq(t,x) =2
et+1ext=∞/summationdisplay
n=0En(x)tn
n!,(see [2, 3, 19-23]) ,
whereEn(x) are the n-th Euler polynomials. For r∈N, the Euler polynomials of
orderris defined by
(4)/parenleftbigg2
et+1/parenrightbiggr
ext=∞/summationdisplay
n=0E(r)
n(x)tn
n!,for|t|< π.
Now we consider the q-extension of (4).
(5)F(r)
q(t,x) = [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mre[m1+···+mr+x]qt=∞/summationdisplay
n=0E(r)
n,q(x)tn
n!,
whereE(r)
n,q(x) are called the n-thq-Euler polynomials of order r(see [10-15]). From
(5), we can derive
(6) E(r)
n,q(x) =[2]r
q
(1−q)nn/summationdisplay
l=0/parenleftbiggn
l/parenrightbigg(−1)lqlx
(1+ql+1)r.
By (5) and (6), we see that
(7) F(r)
q(t,x) = [2]r
q∞/summationdisplay
m=0/parenleftbiggm+r−1
m/parenrightbigg
(−q)me[m+x]qt.
Thus, we note that lim q→1F(r)
q(t,x) =/parenleftBig
2
et+1/parenrightBigr
ext=/summationtext∞
n=0E(r)
n(x)tn
n!.In the special
casex= 0,E(r)
n,q(=E(r)
n,q(0)) are called the n-thq-Euler numbers of order r. By (5),
(6) and (7), we obtain the following proposition.
Proposition 1. Forr∈N, let
F(r)
q(t,x) = [2]r
q/summationdisplay
m1,...,m r=0(−q)m1+···+mre[m1+···+mr+x]qt=∞/summationdisplay
n=0E(r)
n,q(x)tn
n!.
Then we have
E(r)
n,q(x) =[2]r
q
(1−q)nn/summationdisplay
l=0/parenleftbiggn
l/parenrightbigg(−1)lqlx
(1+ql+1)r= [2]r
q∞/summationdisplay
m=0/parenleftbiggm+r−1
m/parenrightbigg
(−q)m[m+x]n
q.
3From the Mellin transformation of F(r)
q(t,x), we can derive the following equation.
1
Γ(s)/integraldisplay∞
0F(r)
q(−t,x)ts−1dt= [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mr
[m1+···+mr+x]sq
= [2]r
q∞/summationdisplay
m=0/parenleftbiggm+r−1
m/parenrightbigg
(−q)m1
[m+x]sq, (8)
wheres∈C,x/negationslash= 0,−1,−2,.... By (8), we can define the multiple q-zeta function
related to q-Euler polynomials.
Definition 2. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the multiple q-zeta
function related to q-Euler polynomials as
ζq,r(s,x) = [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mr
[m1+···+mr+x]sq.
Note that ζq,r(s,x) is a meromorphic function in whole complex s-plane. From (8),
we also note that
ζq,r(s,x) = [2]r
q∞/summationdisplay
m=0/parenleftbiggm+r−1
m/parenrightbigg
(−q)m1
[m+x]sq.
By Laurent series and the Cauchy residue theorem in (5) and (8 ), we see that
ζq(−n,x) =E(n)
n,q(x),forn∈Z+.
Therefore, we obtain the following theorem.
Theorem 3. Forr∈N,n∈Z+, andx∈Rwithx/negationslash= 0,−1,−2,..., we have
ζq(−n,x) =E(r)
n,q(x).
Letχbe the Dirichlet’s character with conductor f∈Nwithf≡1 (mod 2). Then
the generalized q-Euler polynomial attached to χare considered by
Fq,χ(x) =∞/summationdisplay
n=0En,χ,q(x)tn
n!= [2]q∞/summationdisplay
m=0(−q)mχ(m)e[m+x]qt.
From (3) and (9), we have
En,χ,q(x) =[2]q
[2]qff−1/summationdisplay
a=0(−q)aχ(a)En,qf(x+a
f).
4In the special case x= 0,En,χ,q=En,χ,q(0) are called the n-th generated q-Euler
number attached to χ.
It is known that the generalized Euler polynomials of order rare defined by
(10) (2/summationtextf−1
a=0(−1)aχ(a)eat
eft+1)rext=∞/summationdisplay
n=0E(r)
n,χ(x)tn
n!,
for|t|<π
f.
We consider the q-extension of (10). The generalized q-Euler polynomials of order
rattached to χare defined by
F(r)
q,χ(t,x) = [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mr(r/productdisplay
i=1χ(mi))e[m1+···+mr+x]qt
=∞/summationdisplay
n=0E(r)
n,χ,q(x)tn
n!,(see [14, 15]) . (11)
Note that
lim
q→1F(r)
q,χ(t,x) = (2/summationtextf−1
a=0(−1)aχ(a)eat
eft+1)r.
By (11), we easily see that
E(r)
n,χ,q(x) =[2]r
q
(1−q)nn/summationdisplay
l=0/parenleftbiggn
l/parenrightbigg
(−qx)lf−1/summationdisplay
a1,...,ar=0(r/productdisplay
j=1χ(aj))(−ql+1)/summationtextr
i=1ai
(1+q(l+1)f)r
= [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mr(r/productdisplay
i=1χ(mi))[m1+···+mr+x]n
q.
Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we have
1
Γ(s)/integraldisplay∞
0F(r)
q,χ(−t,x)ts−1dt
= [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mr(/producttextr
i=1χ(mi))
[m1+···+mr+x]sq,(see [15]) . (12)
From (12), we can consider the Dirichlet’s type multiple q-l-function as follows :
Definition 4. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the Dirichlet’s
type multiple q-l-function as
lq(s,x|χ) = [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mr(/producttextr
i=1χ(mi))
[m1+···+mr+x]sq,(see [15]) .
By Laurent series and the Cauchy residue theorem in (11) and ( 12), we obtain the
following theorem.
5Theorem 5. Forn∈Z+, we have
lq(−n,x|χ) =E(r)
n,χ,q(x).
Forh∈Zandr∈N, we consider the extended r-pleq-Euler polynomials.
F(h,r)
q(t,x) = [2]r
q∞/summationdisplay
m1,...,m r=0q/summationtextr
j=1(h−j+1)mj(−1)/summationtextr
j=1mje[m1+···+mr+x]qt
=∞/summationdisplay
n=0E(h,r)
n,q(x)tn
n!. (13)
Note that
lim
q→1F(h,r)
q(t,x) = (2
et+1)rext=∞/summationdisplay
n=0E(r)
n(x)tn
n!.
From (13), we note that
E(h,r)
n,q(x) =[2]r
q
(1−q)nn/summationdisplay
l=0/parenleftbiggn
l/parenrightbigg(−qx)l
(−qh−r+l+1:q)r
= [2]r
q∞/summationdisplay
m=0/parenleftbiggm+r−1
m/parenrightbigg
q(−qh−r+1)m[m+x]n
q. (14)
By (14), we easily see that
(15)F(h,r)
q(t,x) = [2]r
q∞/summationdisplay
m=0/parenleftbiggm+r−1
m/parenrightbigg
q(−qh−r+1)me[m+x]qt,(see [11, 13, 14]) .
Using the Mellin transform for F(h,r)
q(t,x), we have
1
Γ(s)/integraldisplay∞
0F(r)
q(−t,x)ts−1dt
= [2]r
q∞/summationdisplay
m1,...,m r=0(−1)m1+···+mrq/summationtextr
j=1(h−j+1)mj
[m1+···+mr+x]sq,(see [13, 14, 15]) ,(16)
fors∈C,x∈Rwithx/negationslash= 0,−1,−2,.... Now we can define the extended q-zeta
function associated with E(h,r)
n,q(x).
6Definition 6. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the (h, q)-zeta
function as
ζ(h)
q,r(s,x) = [2]r
q∞/summationdisplay
m1,...,m r=0(−1)m1+···+mrq/summationtextr
j=1(h−j+1)mj
[m1+···+mr+x]sq.
Notethat ζ(h)
q,r(s,x)isalsoa meromorphic function inwholecomplex s-plane. From
(16) and (15), we note that
(17) ζ(h)
q,r(s,x) = [2]r
q∞/summationdisplay
m=0/parenleftbiggm+r−1
m/parenrightbigg
q(−qh−j+1)m1
[m+x]sq.
Using the Cauchy residue theorem and Laurent series in (16), we obtain the following
theorem.
Theorem 7. Forn∈Z+, we have
ζ(h)
q,r(−n,x) =E(h,r)
n,q(x).
We consider the extended r-ple generalized q-Euler polynomials as follows :
F(h,r)
q,χ(t,x)
= [2]r
q∞/summationdisplay
m1,...,m r=0q/summationtextr
j=1(h−j+1)mj(−1)/summationtextr
j=1mj(r/productdisplay
j=1χ(mj))e[m1+···+mr+x]qt(18)
=∞/summationdisplay
n=0E(h,r)
n,χ,q(x)tn
n!.
By (18), we see that
E(h,r)
n,χ,q(x) =[2]r
q
(1−q)nf−1/summationdisplay
a1,...,ar=0(−1)/summationtextr
j=1aj(r/productdisplay
j=1χ(aj))n/summationdisplay
l=0/parenleftbiggn
l/parenrightbigg(−1)lqlxq(h−j+l+1)aj
(−q(h−r+l+1)f:qf)r
=[2]r
q
[2]r
qf[f]n
qf−1/summationdisplay
a1,...,ar=0(−1)/summationtextr
j=1aj(r/productdisplay
j=1χ(aj))q/summationtextr
j=1(h−j+1)ajζ(h)
qf,r(−n,x+/summationtextr
j=1aj
f).(19)
Therefore, we obtain the following theorem.
7Theorem 8. Forn∈Z+, we have
E(h,r)
n,χ,q(x)
=[2]r
q
[2]r
qf[f]n
qf−1/summationdisplay
a1,...,ar=0(−1)/summationtextr
j=1aj(r/productdisplay
j=1χ(aj))q/summationtextr
j=1(h−j+1)ajζ(h)
qf,r(−n,x+/summationtextr
j=1aj
f).
From (18), we note that
1
Γ(s)/integraldisplay∞
0F(h,r)
q,χ(−t,x)ts−1dt
= [2]r
q∞/summationdisplay
m1,...,m r=0q/summationtextr
j=1(h−j+1)mj(/producttextr
j=1χ(mj))(−1)m1+···+mr
[m1+···+mr+x]sq, (20)
wheres∈C,x∈Rwithx/negationslash= 0,−1,−2,....
From (20), we define the Dirichlet’s type multiple ( h,q)-l-function associated with
the generalized multiple q-Euler polynomials attached to χ.
Definition 9. Fors∈C,x∈Rwithx/negationslash= 0,−1,−2,..., we define the Dirichlet’s
type multiple q-l-function as follows :
l(h)
q(s,x|χ) = [2]r
q∞/summationdisplay
m1,...,m r=0q/summationtextr
j=1(h−j+1)mj(/producttextr
i=1χ(mi))(−1)m1+···+mr
[m1+···+mr+x]sq.
Note that l(h)
q(s,x|χ) is a meromorphic function in whole complex plane. It is easy
to show that
l(h)
q(s,x|χ)
=[2]r
q
[2]r
qf1
[f]sqf−1/summationdisplay
a1,...,ar=0(−1)/summationtextr
j=1aj(r/productdisplay
j=1χ(aj))q/summationtextr
j=1(h−j+1)ajζ(h)
qf,r(s,x+/summationtextr
j=1aj
f).
By (19) and (20), we obtain the following theorem.
Theorem 10. Forn∈Z+, we have
l(h)
q(−n,x|χ) =E(h,r)
n,χ,q(x).
Finally, we give the q-extension of Barnes’ type multiple Euler polynomials in (2 ).
Forx,a1,... ,a r∈Cwith positive real part, let us define the Barnes’ type mutipl e
8q-Euler polynomials in Cas follows :
F(r)
q(t,x|a1,... ,a r;b1,... ,b r)
= [2]r
q∞/summationdisplay
m1,...,m r=0(−1)m1+···+mrq(b1+1)m1+···+(br+1)mre[a1m1+···+armr+x]t(21)
=∞/summationdisplay
n=0E(r)
n,q(x|a1,... ,a r;b1,... ,b r)tn
n!,
whereb1,... ,b r∈Z. By (21), we see that
E(r)
n,q(x|a1,... ,a r;b1,... ,b r)
=[2]r
q
(1−q)nn/summationdisplay
l=0/parenleftbiggn
l/parenrightbigg(−1)lqlx
(1+qla1+b1+1)···(1+qlar+br+1)
= [2]r
q∞/summationdisplay
m1,...,m r=0(−1)m1+···+mrq(b1+1)m1+···+(br+1)mr[a1m1+···+armr+x]n
q.
From (21), we note that
1
Γ(s)/integraldisplay∞
0F(r)
q(−t,x|a1,... ,a r;b1,... ,b r)ts−1dt
= [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr
[a1m1+···+armr+x]sq. (22)
By (22), we define the Barnes’ type multiple q-zeta function as follows :
ζq,r(s,x|a1,... ,a r;b1,... ,b r)
= [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr
[a1m1+···+armr+x]sq,
wheres∈C,x∈Rwithx/negationslash= 0,−1,−2,.... By (21), (22) and (23), we obtain the
following theorem.
Theorem 11. Forn∈Z+, we have
ζq,r(s,x|a1,... ,a r;b1,... ,b r) =E(r)
n,q(x|a1,... ,a r;b1,... ,b r).
Letχbe the Dirichlet’s character with conductor f∈Nwithf≡1 (mod 2). Then
the generalized Barnes’ type multiple q-Euler polynomials attached to χare defined
9by
F(r)
q,χ(t,x|a1,... ,a r;b1,... ,b r)
= [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(r/productdisplay
i=1χ(mi))e[a1m1+···+armr+x]qt(24)
=∞/summationdisplay
n=0E(r)
n,χ,q(x|a1,... ,a r;b1,... ,b r)tn
n!,
From (24), we note that
1
Γ(s)/integraldisplay∞
0F(r)
q,χ(−t,x|a1,... ,a r;b1,... ,b r)ts−1dt
= [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(/producttextr
i=1χ(mi))
[a1m1+···+armr+x]sq. (25)
By (25), we can define Barnes’ type multiple q-l-function in C. Fors∈C,x∈Rwith
x/negationslash= 0,−1,−2,..., let us define the Barnes’ type multiple q-l-function as follows :
l(r)
q(s,x|a1,... ,a r;b1,... ,b r)
= [2]r
q∞/summationdisplay
m1,...,m r=0(−q)m1+···+mrqb1m1+···+brmr(/producttextr
i=1χ(mi))
[a1m1+···+armr+x]sq. (26)
Note that l(r)
q(s,x|a1,... ,a r;b1,... ,b r) is a meromorphic function in whole complex
s-plane. By (24), (25) and (26), we easily see that
l(r)
q(−n,x|a1,... ,a r;b1,... ,b r) =E(r)
n,χ,q(x|a1,... ,a r;b1,... ,b r)
forn∈Z+, (see [1-18]).
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[17] T. Kim, Onp-adicq-l-functions and sums of powers , J. Math. Anal. Appl. 329, 1472–1481.
[18] T. Kim, Y. Simsek, Analytic continuation of the multiple Daehee q-l-functions associated
with Daehee numbers , Russ. J. Math. Phys. 15(2008), 58–65.
[19] Y. H. Kim, W. Kim, C. S. Ryoo, On the twisted q-Euler zeta function associated with
twistedq-Euler numbers , Proc. Jangjeon Math. Soc. 12(2009), 93-100.
[20] H.Ozden, I.N.Cangul, Y.Simsek, Remarks on q-Bernoulli numbers associated with Daehee
numbers , Adv. Stud. Contemp. Math. 18(2009), 41-48.
[21] K. Shiratani, S. Yamamoto, On ap-adic interpolation function for the Euler numbers and
its derivatives , Mem. Fac. Sci., Kyushu University Ser. A 39(1985), 113-125.
[22] Y. Simsek, Theorems on twisted L-function and twisted Bernoulli numbers , Advan. Stud.
Contemp. Math. 11(2005), 205–218.
[23] Z. Zhang, Y. Zhang, Summation formulas of q-series by modified Abel’s lemma , Adv. Stud.
Contemp. Math. 17(2008), 119–129.
Taekyun Kim
Division of General Education-Mathematics, Kwangwoon Uni versity,
Seoul 139-701, S. Korea e-mail: tkkim@kw.ac.kr
Young-Hee Kim
Division of General Education-Mathematics,
Kwangwoon University,
Seoul 139-701, S. Korea e-mail: yhkim@kw.ac.kr
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