Mar 7, 2015 - Seo. For the twisted version of Daehee polynomials, In this paper, we .... We define the twisted modified q-Bernoulli numbers by the generating ...
Advanced Studies in Theoretical Physics Vol. 9, 2015, no. 4, 199 - 211 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.5118
On the Twisted Modified q-Daehee Numbers and Polynomials Dongkyu Lim Department of Mathematics Kyungpook National University, Daegu 702-701, S. Korea c 2015 Dongkyu Lim. This is an open access article distributed under the Copyright Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract The p-adic q-integral(or q-Volkenborn integration) was defined by Kim(see [9,10]). From p-adic q-integrals’ equations, we can derive various q-extension of Bernoulli numbers and polynomials(see [1-21]). In [4], D.S.Kim and T.Kim have studied Daehee numbers and polynomials and their applications. For the twisted Daehee numbers and polynomials are investigate in [17]. In [11], Kim-Lee-Mansour-Seo introduced the q-analogue of Daehee numbers and polynomials which are called q-Daehee numbers and polynomials. In [16], Park investigated twisted version of Daehee polynomials as numbers with q-parameter, which related with usual Bernoulli numbers and polynomials. Lim considered in [13], the modified q-Daehee numbers and polynomials which are different from the q-Daehee numbers and polynomials of Kim-Lee-MansourSeo. For the twisted version of Daehee polynomials, In this paper, we give some useful properties and identities of twisted modified q-Daehee numbers and polynomials related with twisted q-Bernoulli numbers and polynomials.
Mathematics Subject Classification: 11B68, 11S40, 11S80 Keywords: Bernoulli polynomial, Modified q-Daehee polynomial, p-adic q-integral
200
1
Dongkyu Lim
Introduction
Let p be a fixed prime number. Throuhout this paper, Zp , Qp and Cp will repectively denote the ring of p-adic rational integers, the field of p-adic rational numbers and the completion s of algebraic closure of Qp . The p-adic norm is defined |p|p = p1 . When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ C, or p-adic number q ∈ Cp . If q ∈ C, one normally 1 assumes that |q| < 1. If q ∈ Cp , then we assume that |q − 1|p < p− p−1 so that q x = exp(x log q) for each x ∈ Zp . Throughout this paper, we use the notation: [x]q =
1 − qx . 1−q
Note that limq→1 [x]q = x for each x ∈ Zp . Let U D(Zp ) be the space of uniformly differentiable function on Zp . For f ∈ U D(Zp ), the p-adic q-integral on Zp is defined by Kim as follows: Z f (x)dµq (x) = lim
Iq (f ) = Zp
1
pN −1
X
f (x)q x
N →∞ [pN ]q x=0
(see [9, 10]).
(1)
As is well known, the Stirling number of the first kind is defined by x(n) = x(x − 1) · · · (x − n + 1) =
n X
S1 (n, l)xl ,
(2)
l=0
and the Stirling number of the first kind is given by the generating function to be ∞ X tl t m S2 (l, m) (e − 1) = m! (see [8]). (3) l! l=m Unsigned Stirling numbers of the first kind are given by n
x = x(x + 1) · · · (x + n − 1) =
n X
|S1 (n, l)|xl .
(4)
l=0
Note that if we place x to −x in (2), then (−x)(n) = (−1)n xn =
n X
S1 (n, l)(−1)l xl
l=0
= (−1)n
n X l=0
(5) |S1 (n, l)|xl .
201
On the twisted modified q-Daehee numbers and polynomials
Hence, S1 (n, l) = |S1 (n, l)|(−1)n−l . Using integration (1), the q-Daehee polynomials Dn,q (x) are defined and studied by Kim et al.(see [11]), the generating function to be 1−q+
1−q log q
log(1 + t)
1 − q − qt
x
(1 + t) =
∞ X
Dn,q (x)
n=0
tn . n!
(6)
And the modified q-Daehee polynomials are defined and studied by the author. The generating function to be ∞
X tn q − 1 log(1 + t) (1 + t)x = Dn (x|q) log q t n! n=0
(see [13]).
(7)
From (1), we have the following integral identity. qIq (f1 ) − Iq (f ) =
q−1 0 f (0) + (q − 1)f (0), log q
(8)
d where f1 (x) = f (x + 1) and f 0 (x) = dx f (x). tx In special case, we apply f (x) = e on (8), we have the modified q-Bernoulli number Bn (q) as follows: Z ∞ X tn q−1 t −x xt B (q) = (see [13]). (9) q e dµq (x) = n log q et − 1 n=0 n! Zp
Indeed if q → 1, we have lim Bn (q) = Bn . The nth modified q-Bernoulli q→1
polynomials and the generating function to be ∞
X q−1 t tn xt B (x|q) e = . n log q et − 1 n! n=0
(10)
When x = 0, Bn (0|q) = Bn (q) are the nth q-Bernoulli numbers(see [13]). From (8) and (10), we have Z Bn (x|q) = q −y (x + y)n dµq (y). Zp
and Bn (x|q) =
n X n l=0
l
Bl (q)xn−l .
We define the twisted modified q-Bernoulli numbers by the generating function as follows: ∞ X q − 1 ξt tn , (11) Bn,ξ (x) = ξt − 1 n! log q e n=0
202
Dongkyu Lim 1
where |t|p ≤ p− p−1 . If we apply f (x) = q −x eξtx in (8), we have Z q
−x ξtx
e
dµq (x) =
Zp
∞ X
Bn,ξ (q)
n=0
tn . n!
(12)
The nth twisted modified q-Bernoulli polynomials Bn,ξ (x|q) are given by, Z q − 1 ξt ext . (13) Bn,ξ (x|q) = q −x ξ n xn dµq (x) = ξt − 1 log q e Zp The generating function of Daehee polynomials are introduced by Kim as follows: ∞ ∞ Z X X tn log(1 + t) x Dn (x) = (x + y)n dµ0 (x) (see [11]). (14) (1 + t) = n! t Z p n=0 n=0 When x = 0, Dn (0) = Dn are called the Daehee numbers. For n ∈ N, let Tp be the p-adic locally constant space defined by [ Tp = Cpn = lim Cpn , n≥1
n→∞
n
where Cpn = {ω|ω p = 1} is the cyclic group of order pn . 1 We assume that q is an indeterminate in Cp with |1 − q|p < p− p−1 . Then we define the q-analog of a falling fractorial sequence as follows: (x)n,q = x(x − q)(x − 2q) · · · (x − (n − 1)q) (n ≥ 1), Note that lim(x)n,q = (x)n =
q→1
n X
(x)0,1 = 1.
S1 (n, l)xl .
l=0
From the view point of a generalization of the midified q-Daehee polynomials, we consider the twisted modified q-Daehee polynomials defined to be ∞ X n=0
Dn,ξ (x|q)
x q − 1 log (1 + qξt) tn = (1 + qξt) q , n! q log q (1 + qξt) 1q − 1 1
where t, q ∈ Cp with |t|p ≤ |q|p p− p−1 and ξ ∈ Tp The p-adic q-integral(or q-Volkenborn integration) was defined by Kim(see [9,10]). From p-adic q-integrals’ equations, we can derive various q-extension of Bernoulli numbers and polynomials(see [1-21]). In [4], D.S.Kim and T.Kim have studied Daehee numbers and polynomials and their applications. For the twisted
On the twisted modified q-Daehee numbers and polynomials
203
Daehee numbers and polynomials are investigate in [17]. In [11], Kim-LeeMansour-Seo introduced the q-analogue of Daehee numbers and polynomials which are called q-Daehee numbers and polynomials. In [16], Park investigated twisted version of Daehee polynomials as numbers with q-parameter, which related with usual Bernoulli numbers and polynomials. Lim considered in [13], the modified q-Daehee numbers and polynomials which are different from the q-Daehee numbers and polynomials of Kim-Lee-Mansour-Seo. For the twisted version of Daehee polynomials, In this paper, we give some useful properties and identities of twisted modified q-Daehee numbers and polynomials related with twisted q-Bernoulli numbers and polynomials.
2
Witt-type formula for the nth twisted modified q-Daehee polynomials
Let us now consider the p-adic q-integral representation as follows: Z n ξ q −y (x + y)n,q dµq (y) (n ∈ Z+ = N ∪ {0}, ξ ∈ Tp ).
(15)
Zp
From (15), we have n X Z Z ∞ ∞ X tn t n n −y x + y n −y ξ q q = dµq (y) ξ q (x + y)n,q dµq (y) n! n=0 q n! Zp Zp n n=0 Z x+y = q −y (1 + qξt) q dµq (y), Zp
(16) 1 − p−1
where t ∈ Cp with |t|p < |q|p p . x+y 1 − p−1 , we apply f (y) = q −y (1 + qξt) q in (1). For |t|p < |q|p p By (8), we have Z x+y x q − 1 log (1 + qξt) q −y (1 + qξt) q dµq (y) = (1 + qξt) q q log q (1 + qξt) 1q − 1 Zp ∞ X
tn = Dn,ξ (x|q) . n! n=0
(17)
By (16) and (17), we obtain the following theorem, which may be called Witt-type formula for the twisted modified q-Daehee polynomials. Theorem 2.1 For n ≥ 0, we have Z n Dn,ξ (x|q) = ξ q −y (x + y)n,q dµq (y). Zp
204
Dongkyu Lim 1 (eξqt ξq
In (17), by replacing t by
− 1), we have
∞ X
∞
X 1 (eξqt − 1)n q − 1 ξt tn ξxt Dn,ξ (x|q) n n = e = B (x|q) . n,ξ ξ q n! log q eξt − 1 n! n=0 n=0
(18)
and ∞ ∞ ∞ X X Dn,ξ (x|q) 1 ξqt Dn,ξ (x|q) X m m tm n (e − 1) = ξ q S (m, n) 2 ξ n q n n! ξ n q n m=n m! n=0 n=0
=
∞ X m X Dn,ξ (x|q) m=0 n=0
ξ nqn
tm ξ q S2 (m, n) . m!
(19)
m m
By (18) and (19), we obtain the following corollary. Corollary 2.2 For n ≥ 0, we have Bn,ξ (x|q) =
n X
Dm,ξ (x|q)ξ n−m q n−m S2 (n, m).
m=0
By Theorem 2.1, Dn,ξ (x|q) = ξ
n
Z
q −y (x + y)n,q dµq (y)
Zp
Z n X 1 S (n, l) q −y (x + y)l dµq (y). =ξ q l 1 q Z p l=0
(20)
n n
By (20), we obtain the following corollary. Corollary 2.3 For n ≥ 0, we have Dn,ξ (x|q) =
n X
ξ
n−l n−l
q
S1 (n, l)Bl,ξ (x|q) =
l=0
n X
ξ n−l |S1 (n, l)|(−q)n−l Bl,ξ (x|q).
l=0
From now on, we consider twisted modified q-Daehee polynomials of order k ∈ N. Twisted modified q-Daehee polynomials of order k ∈ N are defined by the multivariant p-adic q-integral on Zp : Z Z (k) n Dn,ξ (x|q) = ξ ··· q −(x1 +···+xk ) (x1 + · · · + xk + x)n,q dµq (x1 ) · · · dµq (xk ), Zp
Zp
(21) where n is a nonnegative integer and k ∈ N. In the special case, x = 0, (k) (k) Dn,ξ (q) = Dn,ξ (0|q) are called the twisted modified q-Daehee numbers of order k.
205
On the twisted modified q-Daehee numbers and polynomials (k)
From (21), we can derive the generating function of Dn,ξ (x|q) as follows: ∞ X tn (k) Dn,ξ (x|q) n! n=0
=
∞ X
n n
Z
Z ···
ξ q
Zp
n=0 Z
q
−(x1 +···+xk )
x1 +···+xk +x q
n
Zp
Z
dµq (x1 ) · · · dµq (xk )tn
x1 +···+xk +x
q q −(x1 +···+xk ) (1 + qξt) dµq (x1 ) · · · dµq (xk ) Zp Zp Z Z x1 +···+xk x q = (1 + qξt) ··· q −(x1 +···+xk ) (1 + qξt) q dµq (x1 ) · · · dµq (xk )
=
···
Zp
Zp
q − 1 log (1 + qξt) k . q log q (1 + qξt) 1q − 1
x
= (1 + qξt) q
(22) Note that, by (22), Z Z n X S1 (n, m) (k) n n q −(x1 +···+xk ) (x1 + · · · + xk + x)m Dn,ξ (x|q) = ξ q ··· m q Zp Zp m=0 × dµq (x1 ) · · · dµq (xk ). (23) Since Z
Z
ξ n q −(x1 +···+xk ) e(x1 +···+xk +x)t dµq (x1 ) · · · dµq (xk )
··· Zp
Zp
k q − 1 ξt ext = ξt log q e − 1 ∞ X (k) tn Bn,ξ (x|q) , = n! n=0
we can derive easily Z Z (k) Bn,ξ (x|q) = ··· Zp
ξ n q −(x1 +···+xk ) (x1 + · · · + xk + x)n dµq (x1 ) · · · dµq (xk ).
Zp
(24) Thus, by (23) and (24), we have n X S1 (n, m) (k) (k) n Dn,ξ (x|q) = q ξ n−m Bm,ξ (x|q) qm m=0 = =
n X m=0 n X m=0
(k) q n−m ξ n−m S1 (n, m)Bm (x|q)
(k) ξ n−m |S1 (n, m)|(−q)n−m Bm (x|q).
(25)
206
Dongkyu Lim 1 (eξqt qξ
− 1), we get k X ∞ ∞ X (eξqt − 1)n ξt tn (k) (k) ξtx q − 1 Dn,ξ (x|q) n n =e = B (x|q) . n,ξ ξ q n! log q eξt − 1 n! n=0 n=0
In (22), by replacing t by
(26)
and (k) (k) ∞ ∞ ∞ X X Dn,ξ (x|q) 1 ξqt Dn,ξ (x|q) X tm n m m (e − 1) = ξ q S (m, n) 2 ξ n q n n! ξ n q n m=n m! n=0 n=0 m (k) m ∞ X X Dn,ξ (x|q) t m m . = ξ q S (m, n) 2 nqn ξ m! n=0 m=0
(27)
By (25),(26) and (27), we obtain the following theorem. Theorem 2.4 For n ≥ 0 and k ∈ N, we have (k)
Dn,ξ (x|q) = =
n X m=0 n X
(k) q n−m ξ n−m S1 (n, m)Bm (x|q)
(k) ξ n−m |S1 (n, m)|(−q)n−m Bm (x|q).
m=0
Now, we consider the twisted modified q-Daehee polynomials of the second kind as follows: Z n b Dn,ξ (x|q) = ξ q −y (−y + x)n,q dµq (y) (n ≥ 0). (28) Zp
b n,ξ (q) = D b n,ξ (0|q) are the called the twisted In the special case x = 0, D modified q-Daehee numbers of the second kind. By (28), we have Z −y + x n n b Dn,ξ (x|q) = ξ q q −y dµq (y), (29) n q Zp b n,ξ (x|q) by (8) as follows: and so we can derive the generating function of D Z ∞ ∞ n X X t −y + x tn n n b n,ξ (x|q) = dµ (y) D ξ q q −y q n n! n=0 q n! Zp n=0 −y+x Z ∞ X n n −y q = dµq (y)tn ξ q q n Zp n=0 (30) Z −y+x = q −y (1 + qξt) q dµq (y) Zp x
= (1 + qξt) q
1 − q log (1 + qξt) . q log q (1 + qξt)− 1q − 1
On the twisted modified q-Daehee numbers and polynomials
207
From (29), we get −y + x dµq (y) n q Zp Z n X S1 (n, l) −y n n q =ξ q (−y + x)l dµq (y) l q Zp l=0 Z n X l = ξ l q −y (y − x)l dµq (y)q n−l ξ n−l S1 (n, l)(−1)
b n,ξ (x|q) = ξ n q n D
=
l=0 n X
Z
q −y
(31)
Zp
S1 (n, l)(−1)l Bl,ξ (−x|q)q n−l ξ n−l
l=0
= (−1)n
n X
|S1 (n, l)|Bl,ξ (−x|q)q n−l ξ n−l .
l=0
It is easy to show Bn,ξ (−x|q) = (−1)n Bn,ξ (x + 1|q). Thus from (31), we have n X n b Dn,ξ (x|q) = (−1) |S1 (n, l)|Bl,ξ (−x|q)q n−l ξ n−l l=0
=
n X
(32)
|S1 (n, l)|Bl,ξ (x + 1|q)(−q)n−l ξ n−l .
l=0
From (31) and (32), we have Bn,ξ (x + 1|q) =
n X
b m,ξ (x|q)|S1 (n, m). q n−m ξ n−m D
(33)
m=0
Thus, from (31), we have the following theorem. Theorem 2.5 For n ≥ 0, we have b n,ξ (x|q) = (−1)n D
n X
|S1 (n, l)|Bl,ξ (−x|q)q n−l ξ n−l .
l=0
By replacing t by
1 (eξqt−1 ) qξ
in (30), we have
∞ X
∞
ξqt n X tn b n,ξ (x|q) 1 (e − 1) = q − 1 −ξt eξxt = D . (34) B (−x|q) n,−ξ ξ nqn n! log q e−ξt − 1 n! n=0 n=0
and ∞ b ∞ b ∞ X X Dn,ξ (x|q) 1 ξqt Dn,ξ (x|q) X m m tm n (e − 1) = ξ q S (m, n) 2 ξ n q n n! ξ n q n m=n m! n=0 n=0 =
∞ X m X m=0
m b n,ξ (x|q)S2 (m, n)q m−n ξ m−n t . D m! n=0
(35)
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Dongkyu Lim
By (34) and (35), we obtain the following theorem. Theorem 2.6 For n ≥ 0, we have Bn,−ξ (−x|q) =
n X
b m,ξ (x|q)S2 (n, m). q n−m ξ n−m D
m=0
Now we consider higher-order twisted modified q-Daehee polynomials of the second kind. Higher-order twisted modified q-Daehee polynomials of the second kind are defined by the multivariant p-adic q-integral on Zp : Z Z (k) n b Dn,ξ (x|q) = ξ ··· q −(x1 +···+xk ) (−x1 − · · · − xk + x)n,q dµq (x1 ) · · · dµq (xk ), Zp
Zp
(36) (k) (k) b b where n ∈ Z+ and k ∈ N. In the special case, x = 0, Dn,ξ (q) = Dn,ξ (0|q) are called the higher-order twisted modified q-Daehee numbers of the second kind. b (k) (x|q) as follows: From (36), we can derive the generating function of D n,ξ ∞ n X b (k) (x|q) t D n,ξ n! n=0
=
∞ X
···
ξ q
Zp
n=0
Z
Z
Zp
q Zp
−(x1 +···+xk )
−x1 −···−xk +x q dµq (x1 ) · · · dµq (xk )tn n
q −(x1 +···+xk ) (1 + qξt)
···
=
Z
Z
n n
−x1 −···−xk +x q
dµq (x1 ) · · · dµq (xk )
Zp x
= (1 + qξt) q
1 − q log (1 + qξt) k . q log q (1 + qξt)− 1q − 1 (37)
By (37) b (k) (x|q) = ξ n q n D n,ξ
Z Z n X S1 (n, m) ··· q −(x1 +···+xk ) (−x1 − · · · − xk + x)m m q Zp Zp m=0
× dµq (x1 ) · · · dµq (xk ) Z Z n X S1 (n, m) n n =ξ q ··· q −(x1 +···+xk ) (x1 + · · · + xk + x)m m (−q) Zp Zp m=0 × dµq (x1 ) · · · dµq (xk ) n X S1 (n, m) (k) Bm (−x|q) = ξ nqn m q m=0 n n
=ξ q
n X
(k) q n−m |S1 (n, m)|Bm (−x|q).
m=0
(38)
209
On the twisted modified q-Daehee numbers and polynomials (k)
(k)
It is easy to show Bn (−x|q) = (−1)n Bn (x + k|q). Hence, by (38), Theorem 2.7 For n ≥ 0 and k ∈ N, we have b (k) (x|q) = ξ n q n D n,ξ = ξ nqn
n X
(k)
ξ n−m q n−m |S1 (n, m)|Bm,ξ (−x|q)
m=0 n X
(k)
(−1)m ξ n−m q n−m |S1 (n, m)|Bm,ξ (x + k|q).
m=0
In (37), by replacing t by ∞ X n=0
ξqt n b (k) (x|q) (e − 1) D n,ξ ξ n q n n!
1 (eξqt qξ
=e
ξt(x+k)
− 1), we get
q − 1 ξt log q eξt − 1
k =
∞ X
(k)
Bn,ξ (x + k|q)
n=0
tn . n! (39)
and ∞ b (k) ∞ b (k) ∞ X X Dn,ξ (x|q) X Dn,ξ (x|q) 1 ξqt tm n m m (e − 1) = ξ q S (m, n) 2 ξ n q n n! ξ n q n m=n m! n=0 n=0 m ∞ m b (k) X X Dn,ξ (x|q) t m m = ξ q S2 (m, n) . n n ξ q m! m=0 n=0
(40)
By (39) and (40), we obtain the following theorem. Theorem 2.8 For n ≥ 0 and k ∈ N, we have (k) Bn,ξ (x
+ k|q) =
n X
b (k) (x)ξ n−m q n−m S2 (n, m). D m,ξ
m=0
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