On the multiple q-Genocchi and Euler numbers

0.13 MB·English

by  Taekyun Kim

ON THE MULTIPLE q-GENOCCHI AND EULER NUMBERS
Taekyun Kim
8
0
0
2 Abstract. Thepurposeofthispaperistopresentasystemicstudyofsomefamiliesof
multipleq-GenocchiandEulernumbersbyusingmultivariateq-Volkenbornintegral(=
n
a p-adicq-integral)onZp. Fromthestudiesofthoseq-Genocchinumbersandpolynomials
J of higherorderwe derivesome interestingidentitiesrelatedto q-Genocchinumbersand

7 polynomials of higher order.

]
T
N
§1. Introduction
.
h
t
a
m Let p be a fixed odd prime. Throughout this paper Z , Q , C, and C will,
p p p
[ respectively, denote the ring of p-adic rational integers, the field of p-adic rational

1 numbers, the complex number field, and the completion of algebraic closure of Qp.
v Let v be the normalized exponential valuation of C with |p| = p−vp(p) = p−1
p p p
8
and let q be regarded as either a complex number q ∈ C or a p-adic number q ∈
7
9 C . If q ∈ C, then we always assume |q| < 1. If q ∈ C , we normally assume
p p
0
1. |1−q|p < p−p−11, which implies that qx = exp(xlogq) for |x|p ≤ 1. Here, |·|p is the
0 p-adic absolute value in Cp with |p|p = p1. The q-basic natural number are defined
8 by [n] = 1−qn = 1 + q + ··· + qn−1, ( n ∈ N), and q-factorial are also defined as
0 q 1−q
: [n] ! = [n] ·[n−1] ···[2] ·[1] .In thispaperweuse the notationofGaussian binomial
v q q q q q
i coefficient as follows:
X
r n [n] ! [n] ·[n−1] ···[n−k +1]
a (1) = q = q q q.
(cid:18)k(cid:19) [n−k] ![k] ! [k] !
q q q q
Note that lim n = n = n·(n−1)···(n−k+1). The Gaussian coefficient satisfies the
q→1 k q k n!
following recursion(cid:0) f(cid:1)ormu(cid:0)la(cid:1):
n+1 n n n n
(2) = +qk = qn−k + , cf. [12].
(cid:18) k (cid:19) (cid:18)k −1(cid:19) (cid:18)k(cid:19) (cid:18)k −1(cid:19) (cid:18)k(cid:19)
q q q q q
Key wordsand phrases. q-Bernoullinumbers,q-Volkenbornintegrals,q-Eulernumbers,q-Stirling
numbers.
2000 AMS Subject Classification: 11B68, 11S80
ThispaperissupportedbyJangjeonResearchInstituteforMathematicalScience(JRIMS-10R-2001)
Typeset by AMS-TEX
1
From thus recursion formula we derive
n
= qd1+2d2+···+kdk, cf.[1, 2, 12, 13, 14].
(cid:18)k(cid:19)
q d0+···+dkX=n−k,di∈N
The q-binomial formulae are known as
n n
n
(b;q)n = 1−bqi−1 = q(k2)(−1)kbk,
(cid:18)k(cid:19)
iY=1(cid:0) (cid:1) Xk=0 q
and
n ∞
(3) 1 = 1−bqi−1 −1 = n+k −1 bk, cf.[12].
(b;q) (cid:18) k (cid:19)
n iY=1(cid:0) (cid:1) kX=0 q
In this paper we use the notation
1−qx 1−(−q)x
[x] = , and [x] = .
q −q
1−q 1+q
Hence, lim [x] = 1, for any x with |x| ≤ 1 in the present p-adic case, cf.[1-18].
q→1 q p
For d a fixed positive integer with (p,d) = 1, let
X = X = limZ/dpNZ, X = Z ,
d ←− 1 p
N
X∗ = ∪ (a+dpZ ),
p
0<a<dp
(a,p)=1
a+dpNZ = {x ∈ X|x ≡ a (mod dpn)},
p
where a ∈ Z lies in 0 ≤ a < dpN. In [9], we note that
(−1)aqa (−q)a
µ (a+dpNZ ) = (1+q) = ,
−q p 1+qdpN [dpN]−q
is distribution on X for q ∈ C with |1 − q| < 1. We say that f is a uniformly
p p
differentiable function at a point a ∈ Z and denote this property by f ∈ UD(Z ),
p p
f(x)−f(y)
′
if the difference quotients F (x,y) = have a limit l = f (a) as (x,y) →
f
x−y
(a,a). For f ∈ UD(Z ), this distribution yields an integral as follows:
p
dpN−1
1
I = f(x)dµ (x) = f(x)dµ (x) = lim f(x)(−q)x,
−q ZZp −q ZX −q N→∞ [dpN]−q xX=0
2
which has a sense as we see readily that the limit is convergent (see [9, 10, 14, 15]).
Let q = 1. Then we have the fermionic p-adic integral on Z as follows:
p
pN−1
I = f(x)dµ (x) = lim f(x)(−1)x, cf.[3, 6, 7, 8, 9, 13, 14].
−1 −1
ZZp N→∞ xX=0
For any positive integer N, we set
qa
µ (a+lpNZ ) = , cf. [5, 9, 15, 16, 17, 18],
q p [lpN]
q
and this can be extended to a distribution on X. This distribution yields p-adic
bosonic q-integral as follows (see [12, 17, 18]):
I (f) = f(x)dµ (x) = f(x)dµ (x),
q q q
ZZ ZX
p
where f ∈ UD(Z ) = the space of uniformly differentiable function on Z with val-
p p
ues in C .
p
In view of notation, I can be written symbolically as I (f) = lim I (f),
−1 −1 q→−1 q
cf.[9]. For n ∈ N, let f (x) = f(x+n). Then we have
n
n−1
(4) qnI (f ) = (−1)nI (f)+[2] (−1)n−1−lqlf(l), see [9].
−q n −q q
Xl=0
For any complex number z, it is well known that the familiar Euler polynomialsE (z)
n
are defined by means of the following generating function:
∞
2 tn
F(z,t) = ezt = E (z) , for |t| < π, cf.[13,14].
et +1 n n!
nX=0
We note that, by substituting z = 0, E (0) = E is the familiar n-th Euler number
n n
defined by
∞
2 tn
F(t) = F(0,t) = = E , cf.[12].
et +1 nn!
nX=0
The Genocchi numbers G are defined by the generating function
n
∞
2t tn
= G ,(|t| < π).
et +1 nn!
nX=0
It satisfies G = 1,G = G = ··· = G = 0, and even coefficients are given by
1 3 5 2k+1
G = 2(1−2n)B = 2nE (0),
n n 2n−1
3
where B are Bernoulli numbers and E (x) are Euler polynomials. By meaning of
n n
the generalization of E , Frobenius-Euler numbers and polynomials are also defined
n
by
∞ ∞
1−u tn 1−u tn
= H (u) , and ext = H (u,x) , for u ∈ C, cf.[12, 14].
n n
et −u n! et −u n!
nX=0 nX=0
Over five decades ago, Carlitz [1, 2] defined q-extension of Frobenius-Euler num-
bers and polynomials and proved properties analogous to those satisfied H (u) and
n
H (u,x). In previous my paper [6, 7, 8] the author defined the q-extension of or-
n
dinary Euler and polynomials and proved properties analogous to those satisfied E
n
and E (x). In [6] author also constructed the q-Euler numbers and polynomials of
n
higher order and gave some interesting formulae related to Euler numbers and poly-
nomials of higher order . The purpose of this paper is to present a systemic study
of some families of multiple q-Genocchi and Euler numbers by using multivariate q-
Volkenborn integral (= p-adic q-integral) on Z . From the studies of these q-Genocchi
p
numbers and polynomials of higher order we derive some interesting identities related
to q-Genocchi numbers and polynomials of higher order.
§2. Preliminaries / q-Euler polynomials
− 1
In this section we assume that q ∈ Cp with |1 − q|p < p p−1. Let f1(x) be
translation with f (x) = f(x+1). From (4) we can derive
1
I (f ) = I (f)+2f(0).
−1 1 −1
If we take f(x) = e(x+y)t, then we have Euler polynomials from the integral equation
of I (f) as follows:
−1
∞
2 E (x)tn
e(x+y)tdµ (y) = ext = n .
ZZp −1 et +1 nX=0 n!
That is,
yndµ (y) = E , and (x+y)ndµ (y) = E (x).
−1 n −1 n
ZZ ZZ
p p
Now we consider the following multivariate p-adic fermionic integral on Z as follows:
p
(5)
2 r ∞ tn
ZZp ···ZZp e(x1+···+xr+x)tdµ−1(x1)···dµ−1(xr) = (cid:18)et +1(cid:19) ext = nX=0En(r)(x)n!,
where Er(x) are the Euler polynomials of order r.
n
4
From (5) we note that
(6) ··· (x +···+x +x)ndµ (x )···dµ (x ) = E(r)(x).
1 r −1 1 −1 r n
ZZ ZZ
p p
In view of (6) we can define the q-extension of Euler polynomials of higher order.
For h ∈ Z, k ∈ N, let us consider the extended higher order q-Euler polynomials as
follows:
Em(h,,qk)(x) = ··· [x1+···+xk+x]mq qPkj=1xj(h−j)dµ−q(x1)···dµ−q(xr), see [6] .
ZZ ZZ
p p
From this definition we can derive
m
1 m 1
(7) E(h,k)(x) = [2]k (−1)jqxj , see [6] .
m,q q(1−q)m (cid:18)j (cid:19) (−qj+h;q−1)
Xj=0 k
It is easy to see that
k−1
(−qj+k−1;q−1) = (1+qlqj) = (−qj;q)
k k
Yl=0
In the special case h = k −1, we can easily see that
m
1 m 1
E(k−1.k)(x) = [2]k (−1)jqxj
m,q q(1−q)m (cid:18)j (cid:19) (−qj+k−1;q−1)
Xj=0 k
m
1 m 1
= [2]k (−1)jqxj
q(1−q)m (cid:18)j (cid:19) (−qj;q)
Xj=0 k
(8)
m ∞
1 m k +n−1
= [2]k (−1)jqxj (−qj)n
q(1−q)m (cid:18)j (cid:19) (cid:18) n (cid:19)
Xj=0 nX=0 q
∞
k +n−1
= [2]k (−1)n[n+x]m.
q (cid:18) n (cid:19) q
nX=0 q
Let Fk(t,x) = ∞ E(k−1,k)(x) be the generating function of Ek−1,k(x). From (8)
n=0 n,q n,q
we note that P
∞ ∞ ∞
tm k +n−1 tm
Fk(t,x) = E(k−1,k)(x) = [2]k (−1)n[n+x]m
q m,q m! q (cid:18) n (cid:19) q m!
mX=0 mX=0nX=0 q
∞
k +n−1
= [2]k (−1)ne[n+x]q.
q (cid:18) n (cid:19)
nX=0 q
5
Remark. For w ∈ C with |1−w| < 1, we have
p
∞
2 tn
I (wxetx) = wxetxdµ (x) = = E (w) , see [3, 9, 11, 12, 15] .
−1 ZZp −1 wet +1 nX=0 n n!
Thus, we note that wxxndµ (x) = E (w) and E (w) = 2 H (−w−1), where
Zp −1 n n w+1 n
H (−w−1) are FrobeRnius-Euler numbers.
n
In the previous paper [11], the q-extension of E (w)(= twisted q-Euler numbers)
n
are studied as follows:
∞
tn
(9) I−q(wxet[x]q) = wxet[x]dµ−q(x) = En,q(w) .
ZZp nX=0 n!
From (9) we note that
n
[2] n 1
E (w) = wx[x]ndµ (x) = q (−1)j , see [11] .
n,q ZZp q −q (1−q)n Xj=0(cid:18)j(cid:19) 1+qj+1w
By the exactly same method of Eq.(7), we can also derive the multiple twisted q-Euler
numbers as follows:
E(h,k)(w,x)
m,q
(10)
= ··· wPkj=1xj[x1 +···+xk +x]mq qPkj=1(h−j)xjdµ−q(x1)···dµ−q(xk).
ZZ ZZ
p p
From (10) we can easily derive
[2]k m m 1
E(h,k)(w,x) = q (−qx)l .
m,q (1−q)m (cid:18) l (cid:19) (−wqh+l;q−1)
Xl=0
For h = k −1, we have
[2]k m m 1
E(k−1,k)(w,x) = q (−qx)l
m,q (1−q)m (cid:18) l (cid:19) (−wql;q)
Xl=0 k
(11)
∞
n+k −1
= [2]k (−w)n[n+x]m.
q (cid:18) n (cid:19) q
nX=0 q
Let Fk(w,x) = ∞ E(k−1,k)(x)tm. From (11), we can easily derive
q m=0 m,q m!
P
∞
n+k −1
Fk(w,x) = [2]k (−w)ne[n+x]qt.
q q (cid:18) n (cid:19)
nX=0 q
6
Remark. When we consider those q-Euler numbers and polynomials in complex num-
ber field, we assume that q ∈ C with |q| < 1.
§3. Genocchi and q-Genocchi numbers
From (4) we note that
∞
2t tn
(12) t extdµ (x) = = G .
ZZp −1 et +1 nX=0 nn!
Thus, we have
∞
G tn
(13) extdµ (x) = n+1 .
−1
ZZp nX=0 n+1 n!
By (13) we easily see that
G G (x)
xndµ (x) = n+1, and (x+y)ndµ dµ (y) = n+1 ,
−1 −1 −1
ZZ n+1 ZZ n+1
p p
where G (x) are Genocchi polynomials ( see [8] ).
n
Fromthe multivariatep-adic fermionicintegral onZ we canalso derive theGenoc-
p
chi numbers of order r as follows:
(14)
2t r ∞ tn
trZZp ···ZZp e(x1+···+xr)tdµ−1(x1)···dµ−1(xr) = (cid:18)et +1(cid:19) = nX=0G(nr)n!, r ∈ N,
(r)
where G are the Genocchi numbers of order r.
n
From (14) we note that
∞ ∞
(r+n) tn+r tn
(15) ··· (x +···+x )ndµ (x )···dµ (x ) r = G(r) ,
nX=0ZZp ZZp 1 r −1 1 −1 r (n+r)! nX=0 n n!
where (x) = x(x−1)···(x−r +1). By (14) and (15), we easily see that
r
(16)
1
··· (x +···+x )ndµ (x )···dµ (x ) = G(r) , where n ∈ N∪{0},
ZZ ZZ 1 r −1 1 −1 r r! n+r n+r
p p r
(cid:0) (cid:1)
(r) (r) (r)
and G = G = ··· = G = 0. Thus, we obtain the following theorem.
0 1 r−1
7
Theorem 1. For n ∈ N∪{0}, r ∈ N, we have
G(r) = (n+r) ··· (x +···+x )ndµ (x )···dµ (x )
n+r r 1 r −1 1 −1 r
ZZ ZZ
p p
n+r
= r! ··· (x +···+x )ndµ (x )···dµ (x ),
1 r −1 1 −1 r
(cid:18) r (cid:19) ZZ ZZ
p p
where (x) = x(x−1)···(x−r+1).
r
Recently we constructed the q-extension of Genocchi numbers as follows:
∞ ∞
tn
(17) t e[x]qtdµ−q(x) = [2]qt (−1)nqne[n]qt = Gn,q , see [8] .
ZZp nX=0 nX=0 n!
Thus, we note that
[2] n−1 n−1 (−1)l
[x]ndµ (x) = G = n q , see [8] .
ZZp q −q n,q (1−q)n−1 Xl=0 (cid:18) l (cid:19)1+ql+1
In view of (14) we can define the q-extension of Genocchi numbers of higher order.
For h ∈ Z, k ∈ N, let us consider the extended higher order q-Genocchi numbers as
follows:
∞
tn
nX=0G(nh,q,k)n! = tkZZp ···ZZp e[x1+···+xk]qtqPkj=1xj(h−j)dµ−q(x1)···dµ−q(xk)
∞
tn+k
= nX=0ZZp ···ZZp[x1 +···+xk]nqqPkj=1xj(h−j)dµ−q(x1)···dµ−q(xk) n!
∞
(n+k) tn+k
= nX=0ZZp ···ZZp[x1 +···+xk]nqqPkj=1xj(h−j)dµ−q(x1)···dµ−q(xk) (n+kk)! .
Thus, we have
n+k
G(nh+,kk),q = k!(cid:18) k (cid:19)ZZ ···ZZ [x1 +···+xk]nqqPkj=1xj(h−j)dµ−q(x1)···dµ−q(xk)
p p
n+k [2]k n n 1
= k! q (−1)l ,
(cid:18) k (cid:19)(1−q)n (cid:18)l(cid:19) (−qh+l;q−1)
Xl=0 k
and
(h,k) (h,k) (h,k)
G = G = ··· = G = 0.
0,q 1,q k−1,q
8
If we take h = k −1, then we have
n+k [2]k n n 1
G(k−1,k) = k! q (−1)l
n+k,q (cid:18) k (cid:19)(1−q)n (cid:18)l(cid:19) (−ql;q)
Xl=0 k
n+k [2]k n n ∞ m+k −1
= k! q (−1)l (−ql)m
(cid:18) k (cid:19)(1−q)n (cid:18)l(cid:19) (cid:18) m (cid:19)
Xl=0 mX=0 q
∞
n+k m+k −1
= k! [2]k (−1)m[m]n.
(cid:18) k (cid:19) q (cid:18) m (cid:19) q
mX=0 q
Therefore we obtain the following theorem.
Theorem 2. For h ∈ Z, k ∈ N, we have
n+k [2]k n n 1
G(h,k) = k! q (−1)l ,
n+k,q (cid:18) k (cid:19)(1−q)n (cid:18)l(cid:19) (−qh+l;q−1)
Xl=0 k
and
∞
n+k m+k −1
G(k−1,k) = k! [2]k (−1)m[m]n.
n+k,q (cid:18) k (cid:19) q (cid:18) m (cid:19) q
mX=0 q
Let
∞ ∞
tn tn+k
(18) hk(t) = G(k−1,k) = G(k−1,k) ,
q n,q n! n+k,q (n+k)!
nX=0 nX=0
(k−1,k) (k−1,k)
because G = ··· = G = 0. By (18) and Theorem 2, we see that
0,q k−1,q
∞ ∞ ∞
tn m+k −1 tm
hk(t) = G(k−1,k) = [2]ktk (−1)m[m]n
q n,q n! q (cid:18) m (cid:19) q m!
nX=0 nX=0mX=0 q
∞
m+k −1
= [2]ktk (−1)me[m]qt.
q (cid:18) m (cid:19)
mX=0 q
Remark. For w ∈ C with |1 −w| < 1, we can also define w-Genocchi numbers
p p
(= twisted Genocchi numbers) as follows:
∞
2t tn
t wxextdµ (x) = = G , cf.[3, 11, 13] .
ZZp −1 wet +1 nX=0 n,wn!
From this we note that lim G = G . The q-extension of G can be also
w→1 n,w n n,w
defined by
∞
tn
(19) t wxe[x]qtdµ−q(x) = Gn,q,w , cf. [3, 8, 11, 13] .
ZZp nX=0 n!
9
By (19) we easily see that
[2] n−1 n−1 (−1)l
q
G = n , cf.[11] .
n,q,w (1−q)n−1 (cid:18) l (cid:19)1+ql+1w
Xl=0
From this we also note that lim G = G .
w→1 n,q,w n,q
Now we consider the extended (q,w)-Genocchi numbers by using multivariate p-
adic fermionic integral on Z . For h ∈ Z,k ∈ N, w ∈ C with |1−w| < 1, we define
p p p
(h,k)
G as follows:
n,q,w
∞
tn
G(h,k)
n,q,wn!
(20) nX=0
= tk ··· wx1+···+xke[x1+···+xk]qtqPkj=1xj(h−j)dµ−q(x1)···dµ−q(xk).
ZZ ZZ
p p
From (20) we can derive
n+k [2]k n n 1
G(h,k) = k! q (−1)l ,
n+k,q,w (cid:18) k (cid:19)(1−q)n (cid:18)l(cid:19) (−wqh+l;q−1)
Xl=0 k
and
∞
n+k m+k −1
G(k−1,k) = k! [2]k (−w)m[n]m.
n+k,q,w (cid:18) k (cid:19) q (cid:18) m (cid:19) q
mX=0 q
Let hk (t) = ∞ G(k−1,k)tn. Then we have
q,w n=0 n,q n!
P
∞ ∞
tn+k m+k −1
hk (t) = G(k−1,k) = [2]ktk (−w)me[m]qt.
q,w n+k,q (n+k)! q (cid:18) m (cid:19)
nX=0 mX=0 q
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