New identities involving q-Euler polynomials of higher order

0.13 MB·English

by  Taekyun Kim

NEW IDENTITIES INVOLVING q-EULER
POLYNOMIALS OF HIGHER ORDER
0
1 T. Kim AND Y. H. Kim
0
2
n Abstract. In this paper, we present new generating functions which are related to
a q-Euler numbers and polynomials of higher order. From these generating functions, we
J
give new identities involving q-Euler numbers and polynomials of higher order.

4

]
T
N §1. Introduction/ Preliminaries
.
h
t Let C be the complex number field. We assume that q ∈ C with |q| < 1 and
a
m the q-number is defined by [x] = 1−qx in this paper. The q-factorial is given by
q 1−q
[ [n] ! = [n] [n−1] ···[2] [1] and the q-binomial formulae are known that
q q q q q

1
v n n n
4 (x : q) = (1−xqi−1) = q(2i)(−x)i, (see [3, 14, 15]),
n
3 (cid:18)i(cid:19)
0 iY=1 Xi=0 q
0
and
.
1
0 n ∞
1 1 n+i−1
0 = = xi, (see [3, 5, 14, 15]),
1 (x : q) (cid:18)1−xqi−1(cid:19) (cid:18) i (cid:19)
: n iY=1 Xi=0 q
v
i
X where n = [n]q! = [n]q[n−1]q···[n−i+1]q.
ar The(cid:0)Ei(cid:1)uqler p[no−lyin]qo![mi]qi!als are defin[ie]qd! by 2 ext = ∞ E (x)tn, for |t| < π. In the
et+1 n=0 n n!
special case x = 0, E (= E (0)) are called the n-thPEuler numbers. In this paper, we
n n
consider the q-extensions of Euler numbers and polynomials of higher order. Barnes’
multiple Bernoulli polynomials are also defined by
(1)
tr ∞ tn 2π
ext = B (x,r|a ,··· ,a ) , where |t| < max , (see [1, 14]).
rj=1(eajt −1) nX=0 n 1 r n! 1≤i≤r |ai|
Q
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 Kwangwoon University in 2010
Typeset by AMS-TEX
1
In one of an impressive series of papers (see [1, 6, 14]), Barnes developed the so-called
multiple zeta and multiple gamma function. Let a ,··· ,a be positive parameters.
1 N
Then Barnes’ multiple zeta function is defined by
ζ (s,w|a ,··· ,a ) = (w+m a +···+m a )−s, (see [1]),
N 1 N 1 1 N N
m1,··X·,mN=0
where ℜ(s) > N, ℜ(w) > 0. For m ∈ Z , we have
+
(−1)mm!
ζ (−m,w|a ,··· ,a ) = B (w,N|a ,··· ,a ).
N 1 N N+m 1 N
(N +m)!
In this paper, we consider Barnes’ type multiple q-Euler numbers and polynomials.
The purpose of this paper is to present new generating functions which are related
to q-Euler numbers and polynomials of higher order. From the Mellin transformation
of these generating functions, we derive the q-extensions of 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 ∈ C with |q| < 1. Let x,a ,... ,a be complex
1 r
numbers with positive real parts. Barnes’ type multiple Euler polynomials are defined
by
2r ∞ tn π
(2) ext = E(r)(x|a ,... ,a ) , for |t| < max , (see [6]),
rj=1(eajt +1) nX=0 n 1 r n! 1≤i≤r |wi|
Q
and E(r)(a ,... ,a )(= E(r)(0|a ,... ,a )) are called the n-th Barnes’ type multiple
n 1 r n 1 r
Euler numbers. First, we consider the q-extension of Euler polynomials. The q-Euler
polynomials are defined by
∞ tn ∞
(3) F (t,x) = E (x) = [2] (−q)me[m+x]qt, (see [8, 11, 13, 14, 15]).
q n,q q
n!
nX=0 mX=0
From (3), we have
[2] n n (−1)lqlx
q
E (x) = .
n,q (1−q)n (cid:18)l(cid:19)(1+ql+1)
Xl=0
In the special case x = 0, E (= E (0)) are called the n-th q-Euler numbers. From
n,q n,q
(3), we can easily derive the following relation.
E = 1, and q(qE +1)n +E = 0 if n ≥ 1, (see [8, 16, 17]),
0,q n,q
2
where we use the standard convention about replacing Ek by E . It is easy to show
k,q
that
2 ∞ tn
lim F (t,x) = ext = E (x) , (see [2, 3, 19-23]),
q→1 q et +1 n n!
nX=0
where E (x) are the n-th Euler polynomials. For r ∈ N, the Euler polynomials of
n
order r is defined by
2 r ∞ tn
(4) ext = E(r)(x) , for |t| < π.
(cid:18)et +1(cid:19) n n!
nX=0
Now we consider the q-extension of (4).
∞ ∞ tn
(5) F(r)(t,x) = [2]r (−q)m1+···+mre[m1+···+mr+x]qt = E(r)(x) ,
q q n,q n!
m1,.X..,mr=0 nX=0
where E(r)(x) are called the n-th q-Euler polynomials of order r (see [10-15]). From
n,q
(5), we can derive
[2]r n n (−1)lqlx
(6) E(r)(x) = q .
n,q (1−q)n (cid:18)l(cid:19)(1+ql+1)r
Xl=0
By (5) and (6), we see that
∞
m+r−1
(7) F(r)(t,x) = [2]r (−q)me[m+x]qt.
q q (cid:18) m (cid:19)
mX=0
r
Thus, we note that lim F(r)(t,x) = 2 ext = ∞ E(r)(x)tn. In the special
q→1 q et+1 n=0 n n!
(cid:16) (cid:17)
case x = 0, E(r)(= E(r)(0)) are called the n-th q-EulePr numbers of order r. By (5),
n,q n,q
(6) and (7), we obtain the following proposition.
Proposition 1. For r ∈ N, let
∞ tn
F(r)(t,x) = [2]r (−q)m1+···+mre[m1+···+mr+x]qt = E(r)(x) .
q q n,q n!
m1,.X..,mr=0 nX=0
Then we have
[2]r n n (−1)lqlx ∞ m+r−1
E(r)(x) = q = [2]r (−q)m[m+x]n.
n,q (1−q)n (cid:18)l(cid:19)(1+ql+1)r q (cid:18) m (cid:19) q
Xl=0 mX=0
3
From the Mellin transformation of F(r)(t,x), we can derive the following equation.
q
1 ∞ ∞ (−q)m1+···+mr
F(r)(−t,x)ts−1dt = [2]r
Γ(s) Z q q [m +···+m +x]s
0 m1,.X..,mr=0 1 r q
∞
m+r−1 1
(8) = [2]r (−q)m ,
q (cid:18) m (cid:19) [m+x]s
mX=0 q
where s ∈ C, x 6= 0,−1,−2,.... By (8), we can define the multiple q-zeta function
related to q-Euler polynomials.
Definition 2. For s ∈ C, x ∈ R with x 6= 0,−1,−2,..., we define the multiple q-zeta
function related to q-Euler polynomials as
∞ (−q)m1+···+mr
ζ (s,x) = [2]r .
q,r q [m +···+m +x]s
m1,.X..,mr=0 1 r q
Note that ζ (s,x) is a meromorphic function in whole complex s-plane. From (8),
q,r
we also note that
∞
m+r−1 1
ζ (s,x) = [2]r (−q)m .
q,r q (cid:18) m (cid:19) [m+x]s
mX=0 q
By Laurent series and the Cauchy residue theorem in (5) and (8), we see that
ζ (−n,x) = E(n)(x), for n ∈ Z .
q n,q +
Therefore, we obtain the following theorem.
Theorem 3. For r ∈ N,n ∈ Z , and x ∈ R with x 6= 0,−1,−2,..., we have
+
ζ (−n,x) = E(r)(x).
q n,q
Let χ be the Dirichlet’s character with conductor f ∈ N with f ≡ 1 (mod 2). Then
the generalized q-Euler polynomial attached to χ are considered by
∞ tn ∞
F (x) = E (x) = [2] (−q)mχ(m)e[m+x]qt.
q,χ n,χ,q q
n!
nX=0 mX=0
From (3) and (9), we have
f−1
[2] x+a
E (x) = q (−q)aχ(a)E ( ).
n,χ,q [2] n,qf f
qf aX=0
4
In the special case x = 0, E = E (0) are called the n-th generated q-Euler
n,χ,q n,χ,q
number attached to χ.
It is known that the generalized Euler polynomials of order r are defined by
2 f−1(−1)aχ(a)eat ∞ tn
(10) ( a=0 )rext = E(r)(x) ,
P eft +1 n,χ n!
nX=0
for |t| < π.
f
We consider the q-extension of (10). The generalized q-Euler polynomials of order
r attached to χ are defined by
∞ r
F(r)(t,x) = [2]r (−q)m1+···+mr( χ(m ))e[m1+···+mr+x]qt
q,χ q i
m1,.X..,mr=0 iY=1
∞ tn
(11) = E(r) (x) , (see [14, 15]).
n,χ,q n!
nX=0
Note that
2 f−1(−1)aχ(a)eat
lim F(r)(t,x) = ( a=0 )r.
q→1 q,χ P eft +1
By (11), we easily see that
E(r) (x) = [2]rq n n (−qx)l f−1 ( r χ(a ))(−ql+1)Pri=1ai
n,χ,q (1−q)n (cid:18)l(cid:19) j (1+q(l+1)f)r
Xl=0 a1,.X..,ar=0 jY=1
∞ r
= [2]r (−q)m1+···+mr( χ(m ))[m +···+m +x]n.
q i 1 r q
m1,.X..,mr=0 iY=1
For s ∈ C, x ∈ R with x 6= 0,−1,−2,..., we have
∞
1
F(r)(−t,x)ts−1dt
Γ(s) Z q,χ
0
∞ (−q)m1+···+mr( r χ(m ))
(12) = [2]r i=1 i , (see [15]).
q [m +···+Qm +x]s
m1,.X..,mr=0 1 r q
From (12), we can consider the Dirichlet’s type multiple q-l-function as follows :
Definition 4. For s ∈ C, x ∈ R with x 6= 0,−1,−2,..., we define the Dirichlet’s
type multiple q-l-function as
∞ (−q)m1+···+mr( r χ(m ))
l (s,x|χ) = [2]r i=1 i , (see [15]).
q q [m +···+Qm +x]s
m1,.X..,mr=0 1 r q
By Laurent series and the Cauchy residue theorem in (11) and (12), we obtain the
following theorem.
5
Theorem 5. For n ∈ Z , we have
+
l (−n,x|χ) = E(r) (x).
q n,χ,q
For h ∈ Z and r ∈ N, we consider the extended r-ple q-Euler polynomials.
∞
F(h,r)(t,x) = [2]r qPrj=1(h−j+1)mj(−1)Prj=1mje[m1+···+mr+x]qt
q q
m1,.X..,mr=0
∞ tn
(13) = E(h,r)(x) .
n,q n!
nX=0
Note that
2 ∞ tn
lim F(h,r)(t,x) = ( )rext = E(r)(x) .
q→1 q et +1 n n!
nX=0
From (13), we note that
[2]r n n (−qx)l
E(h,r)(x) = q
n,q (1−q)n (cid:18)l(cid:19)(−qh−r+l+1 : q)
Xl=0 r
∞
m+r−1
(14) = [2]r (−qh−r+1)m[m+x]n.
q (cid:18) m (cid:19) q
mX=0 q
By (14), we easily see that
∞
m+r−1
(15) F(h,r)(t,x) = [2]r (−qh−r+1)me[m+x]qt, (see [11, 13, 14]).
q q (cid:18) m (cid:19)
mX=0 q
Using the Mellin transform for F(h,r)(t,x), we have
q
∞
1
F(r)(−t,x)ts−1dt
Γ(s) Z q
0
(16)
∞ (−1)m1+···+mrqPrj=1(h−j+1)mj
= [2]r , (see [13, 14, 15]),
q [m +···+m +x]s
m1,.X..,mr=0 1 r q
for s ∈ C, x ∈ R with x 6= 0,−1,−2,.... Now we can define the extended q-zeta
function associated with E(h,r)(x).
n,q
6
Definition 6. For s ∈ C, x ∈ R with x 6= 0,−1,−2,..., we define the (h, q)-zeta
function as
∞ (−1)m1+···+mrqPrj=1(h−j+1)mj
ζ(h)(s,x) = [2]r .
q,r q [m +···+m +x]s
m1,.X..,mr=0 1 r q
Notethat ζ(h)(s,x) is alsoa meromorphic function in whole complex s-plane. From
q,r
(16) and (15), we note that
∞
m+r−1 1
(17) ζ(h)(s,x) = [2]r (−qh−j+1)m .
q,r q (cid:18) m (cid:19) [m+x]s
mX=0 q q
Using the Cauchy residue theorem and Laurent series in (16), we obtain the following
theorem.
Theorem 7. For n ∈ Z , we have
+
ζ(h)(−n,x) = E(h,r)(x).
q,r n,q
We consider the extended r-ple generalized q-Euler polynomials as follows :
F(h,r)(t,x)
q,χ
(18)
∞ r
= [2]rq qPrj=1(h−j+1)mj(−1)Prj=1mj( χ(mj))e[m1+···+mr+x]qt
m1,.X..,mr=0 jY=1
∞ tn
= E(h,r)(x) .
n,χ,q n!
nX=0
By (18), we see that
En(h,χ,r,q)(x) = (1[−2]rqq)n f−1 (−1)Prj=1aj( r χ(aj)) n (cid:18)nl(cid:19)((−−1q)(hlq−lrx+q(l+h−1)jf+l:+q1f))aj
a1,.X..,ar=0 jY=1 Xl=0 r
(19)
[2]r f−1 r x+ r a
= [2]rq [f]nq (−1)Prj=1aj( χ(aj))qPrj=1(h−j+1)ajζq(hf,)r(−n, Pfj=1 j).
qf a1,.X..,ar=0 jY=1
Therefore, we obtain the following theorem.
7
Theorem 8. For n ∈ Z , we have
+
E(h,r)(x)
n,χ,q
[2]r f−1 r x+ r a
= [2]rq [f]nq (−1)Prj=1aj( χ(aj))qPrj=1(h−j+1)ajζq(hf,)r(−n, Pfj=1 j).
qf a1,.X..,ar=0 jY=1
From (18), we note that
∞
1
F(h,r)(−t,x)ts−1dt
Γ(s) Z q,χ
0
(20) = [2]r ∞ qPrj=1(h−j+1)mj( rj=1χ(mj))(−1)m1+···+mr,
q [m +Q···+m +x]s
m1,.X..,mr=0 1 r q
where s ∈ C, x ∈ R with x 6= 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. For s ∈ C, x ∈ R with x 6= 0,−1,−2,..., we define the Dirichlet’s
type multiple q-l-function as follows :
l(h)(s,x|χ) = [2]r ∞ qPrj=1(h−j+1)mj( ri=1χ(mi))(−1)m1+···+mr.
q q [m +Q···+m +x]s
m1,.X..,mr=0 1 r q
Note that l(h)(s,x|χ) is a meromorphic function in whole complex plane. It is easy
q
to show that
l(h)(s,x|χ)
q
= [[22]]rrq [f1]s f−1 (−1)Prj=1aj( r χ(aj))qPrj=1(h−j+1)ajζq(hf,)r(s, x+Pfrj=1aj).
qf q a1,.X..,ar=0 jY=1
By (19) and (20), we obtain the following theorem.
Theorem 10. For n ∈ Z , we have
+
l(h)(−n,x|χ) = E(h,r)(x).
q n,χ,q
Finally, we give the q-extension of Barnes’ type multiple Euler polynomials in (2).
For x,a ,... ,a ∈ C with positive real part, let us define the Barnes’ type mutiple
1 r
8
q-Euler polynomials in C as follows :
F(r)(t,x|a ,... ,a ;b ,... ,b )
q 1 r 1 r
(21)
∞
= [2]r (−1)m1+···+mrq(b1+1)m1+···+(br+1)mre[a1m1+···+armr+x]t
q
m1,.X..,mr=0
∞ tn
= E(r)(x|a ,... ,a ;b ,... ,b ) ,
n,q 1 r 1 r n!
nX=0
where b ,... ,b ∈ Z. By (21), we see that
1 r
E(r)(x|a ,... ,a ;b ,... ,b )
n,q 1 r 1 r
[2]r n n (−1)lqlx
q
=
(1−q)n (cid:18)l(cid:19)(1+qla1+b1+1)···(1+qlar+br+1)
Xl=0
∞
= [2]r (−1)m1+···+mrq(b1+1)m1+···+(br+1)mr[a m +···+a m +x]n.
q 1 1 r r q
m1,.X..,mr=0
From (21), we note that
∞
1
F(r)(−t,x|a ,... ,a ;b ,... ,b )ts−1dt
Γ(s) Z q 1 r 1 r
0
∞ (−q)m1+···+mrqb1m1+···+brmr
(22) = [2]r .
q [a m +···+a m +x]s
m1,.X..,mr=0 1 1 r r q
By (22), we define the Barnes’ type multiple q-zeta function as follows :
ζ (s,x|a ,... ,a ;b ,... ,b )
q,r 1 r 1 r
∞ (−q)m1+···+mrqb1m1+···+brmr
= [2]r ,
q [a m +···+a m +x]s
m1,.X..,mr=0 1 1 r r q
where s ∈ C, x ∈ R with x 6= 0,−1,−2,.... By (21), (22) and (23), we obtain the
following theorem.
Theorem 11. For n ∈ Z , we have
+
ζ (s,x|a ,... ,a ;b ,... ,b ) = E(r)(x|a ,... ,a ;b ,... ,b ).
q,r 1 r 1 r n,q 1 r 1 r
Let χ be the Dirichlet’s character with conductor f ∈ N with f ≡ 1 (mod 2). Then
the generalized Barnes’ type multiple q-Euler polynomials attached to χ are defined
9
by
F(r)(t,x|a ,... ,a ;b ,... ,b )
q,χ 1 r 1 r
(24)
∞ r
= [2]r (−q)m1+···+mrqb1m1+···+brmr( χ(m ))e[a1m1+···+armr+x]qt
q i
m1,.X..,mr=0 iY=1
∞ tn
= E(r) (x|a ,... ,a ;b ,... ,b ) ,
n,χ,q 1 r 1 r n!
nX=0
From (24), we note that
∞
1
F(r)(−t,x|a ,... ,a ;b ,... ,b )ts−1dt
Γ(s) Z q,χ 1 r 1 r
0
∞ (−q)m1+···+mrqb1m1+···+brmr( r χ(m ))
(25) = [2]r i=1 i .
q [a m +···+a m +Qx]s
m1,.X..,mr=0 1 1 r r q
By (25), we can define Barnes’ type multiple q-l-function in C. For s ∈ C, x ∈ R with
x 6= 0,−1,−2,..., let us define the Barnes’ type multiple q-l-function as follows :
l(r)(s,x|a ,... ,a ;b ,... ,b )
q 1 r 1 r
∞ (−q)m1+···+mrqb1m1+···+brmr( r χ(m ))
(26) = [2]r i=1 i .
q [a m +···+a m +Qx]s
m1,.X..,mr=0 1 1 r r q
Note that l(r)(s,x|a ,... ,a ;b ,... ,b ) is a meromorphic function in whole complex
q 1 r 1 r
s-plane. By (24), (25) and (26), we easily see that
l(r)(−n,x|a ,... ,a ;b ,... ,b ) = E(r) (x|a ,... ,a ;b ,... ,b )
q 1 r 1 r n,χ,q 1 r 1 r
for n ∈ Z , (see [1-18]).
+
References
[1] E. W. Barnes, On the theory of multiple gamma function, Trans. Camb. Ohilos. Soc. A
196 (1904), 374-425.
[2] I. N. Cangul,V. Kurt, H. Ozden, Y. Simsek, On the higher-order w-q-Genocchi numbers,
Adv. Stud. Contemp. Math. 19 (2009), 39–57.
[3] N. K.Govil, V. Gupta, Convergence of q-Meyer-Konig-Zeller-Durrmeyer operators, Adv.
Stud. Contemp. Math. 19 (2009), 97–108.
[4] T.Kim,Onaq-analogueofthep-adicloggammafunctionsandrelatedintegrals,J.Number
Theory 76 (1999), 320–329.
[5] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002), 288–299.
[6] T.Kim,OnEuler-Barnesmultiplezetafunctions,Russ.J.Math.Phys.10(2003),261–267.
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