New approach to q-Genocch, Euler numbers and polynomials and their interpolation functions

0.11 MB·English

by  Taekyun Kim

NEW APPROACH TO q-EULER, GENOCCHI NUMBERS
AND THEIR INTERPOLATION FUNCTIONS
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Taekyun Kim
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n Abstract. In [1], Cangul-Ozden-Simsek constructed a q-Genocchi numbers of higher
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order and gave Witt’sformula of these numbers by using a p-adic fermionic integral on
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4 Zp. In this paper, we give another constructions of a q-Euler and Genocchi numbers
of higher order, which are different than their q-Genocchi and Euler numbers of higher

] order. By using our q-Euler and Genocchi numbers of higher order, we can investigate
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the interesting relationship between q-w-Euler numbers and q-w-Genocchi numbers.
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Finally, we give the interpolation functions of these numbers.
.
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t
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§1. Introduction/ Preliminaries
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1
v
3 In [1], Cangul-Ozden-Simsek presented a systematic study of some families of the
5 q-Genocchi numbers of higher order. By applying the q-Volkenborn integral in the
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sense of fermionic, they also constructed the q-extension of Euler zeta function, which
0
. interpolatesthesenumbers atnegativeintegers. SomepropertiesofGenocchi numbers
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0 of higher order were treated in [1, 11, 12, 13, 14, 15, 16, 17]. So, we will construct the
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appropriate the q-Genocchi numbers of higher order to be related q-Euler numbers
0
: of higher order for doing study the q-extension of Genocchi numbers of higher order
v
i in this paper. Let p be a fixed odd prime. Throughout this paper Zp, Qp, C and
X
C will, respectively, denote the ring of p-adic rational integers, the field of p-adic
p
r
a rational numbers, the complex number field and the completion of algebraic closure
of Q . Let v be the normalized exponential valuation of C with |p| = p−vp(p) = 1.
p p p p p
The q-basic natiral numbers are defined by [n] = 1−qn = 1+q+q2 +···+qn−1 for
q 1−q
n ∈ N, and the binomial coefficient is defined as
n n! n(n−1)···(n−k +1)
= = .
(cid:18)k(cid:19) (n−k)!k! k!
Key words and phrases. :p-adic q-deformed fermionic integral, Higher order twisted q-Genocchi
and Euler numbers, Bernoulli numbers , interpolations.
2000 AMS Subject Classification: 11B68, 11S80 integral
The present Research has been conducted by the research Grant of Kwangwoon University in 2008
Typeset by AMS-TEX
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The binomial formulas are well known that
n n
n 1 n+i−1
(1−b)n = (−1)ibi, and = bi, (see [1-20]).
(cid:18)i(cid:19) (1−b)n (cid:18) i (cid:19)
Xi=0 Xi=0
In this paper, we use the notation
1−qx 1−(−q)x
[x]q = , and [x]−q = , (see [1-22]).
1−q 1+q
We say that f is uniformly differentiable function at a point a ∈ Z , and write
p
f ∈ UD(Z ), if the difference quotient F (x,y) = f(x)−f(y) have a limit f′(a) as
p f x−y
(x,y) → (a,a). For f ∈ UD(Z ), q ∈ C with |1 − q| < 1, an invariant p-adic
p p p
q-integral is defined as
pN−1
1+q
(1) I−q(f) = ZZp f(x)dµ−q(x) = Nl→im∞ 1+qpN xX=0 f(x)(−q)x, see [9].
Thus, we have the following integral relation:
qI−q(f1)+I−q(f) = (1+q)f(0), where f1(x) = f(x+1) .
The fermionic p-adic invariant integral on Z is defined as
p
I−1(f) = lim I−q(f) = f(x)dµ−1(x).
q→1 ZZ
p
For n ∈ N, let f −n(x) = f(x+n). From (1), we can derive
n−1
qnI−q(fn) = (−1)nI−q(f)+[2]q (−1)n−1−lqlf(l).
Xl=0
It is known that the w-Euler polynomials are defined as
∞
2ext tn
(2) = E (x) , (see [ 20, 21, 22]).
wet +1 n,w n!
nX=0
Note that E (0) = E are called the w-Euler numbers. w-Genocchi polynomials
n,w n,w
are defined as
∞
2t tn
ext = G (x) , (see [1]).
n,w
wet +1 n!
nX=0
2
In the special case x = 0, G (0) = G are called w-Genocchi numbers. In [1],
n,w n,w
Cangul-Ozden-Simsek have studied w-Genocchi numbers of order r as follows. For
w ∈ C with |1−w| < 1, the w-Genocchi numbers of order r are given by
p p
t r ∞ tn
(3) ZZp ···ZZp et(x1+···+xr)dµ−1(x1)···dµ−1(xr) = 2r(cid:18)wet +1(cid:19) = nX=0G(nr,)wn!.
They also considered the q-extension of (3) as follows.
··· qPrv=0(h−v)xvet(x1+···+xr)dµ−1(x1)···dµ−1(xr)
ZZ ZZ
p p
(4) ∞
2rtr tn
= = G(h,r) , (see [1]) .
(qhet +1)(qh−1et +1)···(qh−r+1et +1) n,q n!
nX=0
The purpose of this paper is to give another construction of q-Euler numbers and
q-Genocchi numbers of higher order, which are different than a q-Genocchi numbers
of Cangul-Ozden-Simsek.
§2.New approach to q-Euler, Genocchi numbers and polynomials
In this section, we assume that q ∈ C with |1 − q| < 1 and let w ∈ C with
p p p
(r)
|1−w| < 1. The w-Euler polynomials of order r, denoted E (x), are defined as
p n
2 r 2 2 ∞ tn
ext = ×···× ext = E(r) (x) , (see [1]).
(cid:18)wet +1(cid:19) (cid:18)wet +1(cid:19) (cid:18)wet +1(cid:19) n,w n!
nX=0
r−times
| {z }
(r)
The values of E (x) at x = 0 are called w-Euler number of order r: when r = 1
n,w
and w = 1, the polynomials or numbers are called the ordinary Euler polynomials or
(r) (r)
numbers. When x = 0 or r = 1, we use the following notation: E denote E (0),
n,w n,w
(1) (1)
E (x) denote E (x), and E denote E (0).
n,w n,w n,w n,w
It is known that
∞
2 tn
ZZp wxetxdµ−1(x) = wet +1 = nX=0En,wn!,
and
∞
2 tn
ZZp wyet(x+y)dµ−1(y) = wet +1ext = nX=0En,w(x)n!, (see ([1])
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The higher order w-Euler numbers and polynomials are given by
2 r ∞ tn
ZZp ···ZZp wx1+···+xre(x1+···+xr)tdµ−1(x1)···dµ−1(xr) = (cid:18)wet +1(cid:19) = nX=0En(r,w) n!,
and
(5)
r 2 r ∞ tn
ZZp ···ZZp wPri=1xie(Pri=1xi+x)tiY=1dµ−1(xi) = (cid:18)wet +1(cid:19) ext = nX=0En,w(x)(r)n!.
Thus, we have the following theorem.
Proposition 1. For n ∈ N, we have
(6) En(r,w) (x) = ··· wx1+···+xr(x1 +···+xr +x)ndµ−1(x1)···dµ−1(xr),
ZZ ZZ
p p
and
En(r,w) = ··· wx1+···+xr(x1 +···+xr)ndµ−1(x1)···dµ−1(xr).
ZZ ZZ
p p
From the results of Cangul-Ozden-Simsek (see [1]), we can derive the following
w-Genocchi polynomials of order r as follows.
tr ··· wx1+x2+···+xre(x1+x2+···+xr+x)tdµ−1(x1)dµ−1(x2)···dµ−1(xr)
ZZ ZZ
p p
(7) 2 r ∞ tn
= ext = G(r) (x) , (see [1]).
(cid:18)wet +1(cid:19) n,w n!
nX=0
From (7), we note that
tr ··· wx1+x2+···+xre(x1+x2+···+xr+x)tdµ−1(x1)dµ−1(x2)···dµ−1(xr)
ZZ ZZ
p p
(8) ∞
n+r tn+r
= ··· wx1+···+xr(x +···+x +x)nr! .
1 r
nX=0ZZp ZZp (cid:18) r (cid:19)(n+r)!
(r) (r) (r)
By (6), (7) and (8), we easily see that G (x) = G (x) = ··· = G (x) = 0,
0,w 1,w n−1,w
and
(r)
G (x)
rn!+nr,+wr = ZZ ···ZZ wx1+···+xr(x1 +x2 +···+xr +x)ndµ−1(x1)···dµ−1(xr)
r p p
(cid:0) (cid:1) = E(r) (x).
n,w
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In the viewpoint of the q-extension of (6), let us define the w-q-Euler numbers of
order r as follows.
(9) En(r,w) ,q = ··· wx1+···+xr[x1+···+xr]nqq−(x1+···+xr)dµ−q(x1)···dµ−q(xr).
ZZ ZZ
p p
From (9), we note that
n r
1 n [2]
E(r) = (−1)l q
n,w,q (1−q)n (cid:18)l(cid:19) (cid:18)1+wql(cid:19)
Xl=0
n ∞
1 n m+r −1
(10) = [2]r (−1)l (−1)mwmqlm
q(1−q)n (cid:18)l(cid:19) (cid:18) m (cid:19)
Xl=0 mX=0
∞
m+r −1
= [2]r (−1)mwm[m]n.
q (cid:18) m (cid:19) q
mX=0
Therefore, we obtain the following theorem.
Theorem 2. For n ∈ N, we have
∞
m+r −1
E(r) = [2]r (−1)mwm[m]n.
n,w,q q (cid:18) m (cid:19) q
mX=0
Let F(t,x|q) = ∞ E(r) tn. Then we have
n=0 n,w,qn!
P
F(t,w|q) = ··· wx1+···+xrq−(x1+···+xr)e[x1+···+xr]qtdµ−q(x1)···dµ−q(xr)
ZZ ZZ
p p
∞
m+r −1
= [2]r (−1)mwme[m]qt.
q (cid:18) m (cid:19)
mX=0
Thus, we obtain the following corollary.
Corollary 3. Let F(t,x|q) = ∞ E(r) tn. Then we have
n=0 n,w,qn!
P
∞
m+r −1
F(t,x|q) = [2]r (−1)mwme[m]qt.
q (cid:18) m (cid:19)
mX=0
Note that
∞
dk m+r −1
F(t,x|q) = E(r) = [2]r (−1)mwm[m]k.
dtk t=0 k,w,q q (cid:18) m (cid:19) q
(cid:12) mX=0
(cid:12) 5
Let us define the q-extension of w-Genocchi numbers of order r as follows.
tr ··· q−(x1+···+xr)wx1+···+xre[x1+···+xr]qtdµ−q(x1)···dµ−q(xr)
ZZ ZZ
p p
(11) ∞ ∞
m+r −1 tn
= tr[2]r (−1)mwme[m]qt = G(r) .
q (cid:18) m (cid:19) n,w,qn!
mX=0 nX=0
From (11), we can derive
∞ r r! n+r tn+r
nX=0ZZp ···ZZp q−(x1+···+xr)wx1+···+xr[x1 +···+xr]nq iY=1dµ−q(xi) (cid:0)(nr+(cid:1)r)!
(12)
∞ ∞
tn tn+r
= G(r) = G(r) .
n,w,qn! n+r,w,q(n+r)!
nX=0 nX=0
By (9), (11) and (12), we obtain the following theorem.
Theorem 4. For n ∈ Z , and r ∈ N, we have
+
(r)
G
r!n+nr+,wr,q = ZZ ···ZZ q−(x1+···+xr)wx1+···+xr[x1 +···+xr]nqdµ−q(x1)···dµ−q(xr)
r p p
(cid:0) (cid:1) = E(r) ,
n,w,q
and G(r) = G(r) = ··· = G(r) = 0.
0,w,q 1,w,q r−1,w,q
In the sense of the q-extension of (6), we can consider the w-q-Euler polynomials
of order r as follows.
(13)
En(r,w) ,q(x) = ··· wx1+···+xrq−(x1+···+xr)[x1 +···+xr]nqdµ−q(x1)···dµ−q(xr)
ZZ ZZ
p p
n r
1 n [2]
= (−1)lqlx q
(1−q)n (cid:18)l(cid:19) (cid:18)1+wql(cid:19)
Xl=0
∞
m+r−1
= [2]r (−1)m[m+x]nwm.
q (cid:18) m (cid:19) q
mX=0
Let F(t,w,x|q) = ∞ E(r) (x)tn. Then we have
n=0 n,w,q n!
P
∞
m+r −1
F(t,w,x|q) = [2]r (−1)mwme[m+x]qt
q (cid:18) m (cid:19)
(14) mX=0
= ··· wx1+···+xrq−(x1+···+xr)e[x1+···+xr+x]qtdµ−q(x1)···dµ−q(xr).
ZZ ZZ
p p
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In the viewpoint of (7), we can define the w-q-Genocchi polynomials of order r as
follows.
∞ ∞
tn m+r −1
(15) G(r) (x) = [2]rtr (−1)mwme[m+x]qt.
n,w,q n! (cid:18) m (cid:19)
nX=0 mX=0
By (14) and (15), we obtain the following theorem.
Theorem 5. For n ∈ Z , r ∈ N, we have
+
(r)
G (x)
n+r,w,q = E(r) (x)
r! n+r n,w,q
r
(cid:0) (cid:1)
= ··· wx1+···+xrq−(x1+···+xr)[x1 +···+xr]nqdµ−q(x1)···dµ−q(xr),
ZZ ZZ
p p
and
(r) (r)
G (x) = ··· = G (x) = 0.
0,w,q r−1,w,q
§3. Further Remarks and Observation for the multiple w-q-zeta function
In this section, we assume that q ∈ C with |q| < 1. For s ∈ C, w = e2πiξ(ξ ∈ R),
let us define the Lerch type q-zeta function of order r as follows.
∞ m+r−1 (−1)mwm
(16) ζ(r)(s) = [2]r m .
q,w q (cid:0) [(cid:1)m]s
mX=1 q
For k ∈ N, by (10) and (16), we have ζ(r)(−k) = E(r) .
q,w k,w,q
We now consider the q-extension of Hurwitz-Lerch type zeta function of order r as
follows.
∞ m+r−1 (−1)mwm
ζ(r)(s,x) = [2]r m ,
q,w q (cid:0) [m(cid:1)+x]s
mX=0 q
where x ∈ C with x 6= 0,−1,−2,··· , and s ∈ C.
From (13) and (17), we can also derive
ζ(r)(−k,x) = E(r) (x), for k ∈ N.
q,w n,w,q
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Taekyun Kim
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-
701, S. Korea e-mail: [email protected]
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