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J. Nonlinear Sci. Appl. 9 (2016), 1872–1876
Research Article
On the Appell type λ-Changhee polynomials
Dongkyu Lima,∗, Feng Qib,c
a
School of Mathematical Sciences, Nankai University, Tianjin Ciy, 300071, China.
b
Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China.
c
Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China.
Communicated by C. Park
Abstract
In the paper, by virtue of the p-adic fermionic integral on Zp , the authors consider a λ-analogue of the
c
Changhee polynomials and present some properties and identities of these polynomials. 2016
All rights
reserved.
Keywords: Identity, property, Appell type λ-Changhee polynomial, Changhee polynomial, degenerate
Changhee polynomial.
2010 MSC: 05A30, 05A40, 11B68, 11S80.
1. Introduction
Let p be a fixed odd prime number and let Zp , Qp , and Cp denote respectively the ring of p-adic integers,
the field of p-adic rational numbers, and the completion of the algebraic closure of Qp . The p-adic norm is
normally defined as |p| = p1 . Recently, degenerate Changhee polynomials Chn,λ (x) are defined [6, p. 296] by
∞
2λ
ln(1 + λt) x X
tn
1+
=
Chn,λ (x) .
2λ + ln(1 + λt)
λ
n!
n=0
When x = 0, we call Chn,λ = Chn,λ (0) the degenerate Changhee numbers.
It is common knowledge that the Euler polynomials En (x) are given by
∞
X
2
tn
xt
e
=
E
(x)
n
et + 1
n!
n=0
∗
Corresponding author
Email addresses: dgrim84@gmail.com (Dongkyu Lim), qifeng618@gmail.com, qifeng618@hotmail.com (Feng Qi)
Received 2015-08-18
D. Lim, F. Qi, J. Nonlinear Sci. Appl. 9 (2016), 1872–1876
1873
and that, when x = 0, we call En = 2n En 12 the Euler numbers.
Let C(Zp ) be the space of all continuous functions on Zp . For f ∈ C(Zp ), the fermionic p-adic integral
on Zp was defined [3, p. 134] by
Z
f (x) d µ−1 (x) = lim
I−1 (f ) =
N →∞
Zp
N −1
pX
N
f (x)µ−1 x + p Zp = lim
N →∞
x=0
N −1
pX
f (x)(−1)x .
x=0
From the above definition, we can derive
I−1 (f1 ) + I−1 (f ) = 2f (0),
where f1 (x) = f (x + 1). Consequently, it follows from [2, p. 1256], [4, p. 994], and [5, p. 366] that
Z
∞
X
tn
x+y
(1 + t)
d µ−1 (y) =
Chn (x) ,
n!
Zp
n=0
where Chn (x) are called the Changhee polynomials.
We note that the Euler polynomials En (x) may also be represented by
Z
∞
X
2
tn
(x+y)t
xt
e
d µ−1 (y) = t
e =
En (x) .
e +1
n!
Zp
n=0
The purpose of this paper is to construct a new type of polynomials, the Appell type λ-Changhee
polynomials, and to investigate some properties and identities of these polynomials.
2. Appell type λ-Changhee polynomials
Assume that λ, t ∈ Cp such that
| λt |p < p−1/(p−1)
and
(1 + λt)x/λ = ex ln(1+λt)/λ .
Now we define the Appell type λ-Changhee polynomials Chn (x|λ) by
Z
∞
X
2
tn
xt
ey ln(1+λt)/λ+xt d µ−1 (y) =
Ch
(x|λ)
.
e
=
n
n!
(1 + λt)1/λ + 1
Zp
n=0
When x = 0, we call Chn (λ) = Chn (0|λ) the λ-Changhee numbers. Note that Chn (1) = Chn for n ≥ 0.
Theorem 2.1. For n ≥ 0, we have
n X
n
Chn (x|λ) =
Chm (λ)xn−m .
m
m=0
Proof. From (2.1), we can derive
∞
X
n=0
"∞
# ∞
!
" n #
∞ X
X
X xl
X
tn
n
tn
Chn (x|λ) =
Chn (λ)
tl =
.
Chm (λ)xn−m
n!
l!
m
n!
n=0
l=0
n=0 m=0
Equating coefficients on the very ends of the above identity arrives at the required result.
Theorem 2.2. For n ≥ 0, we have
n X
k X
n k−m
Chn (x|λ) =
λ
Em (0)S1 (k, m)xn−k ,
k
k=0 m=0
where S1 (n, m) is the Stirling number of the first kind.
(2.1)
D. Lim, F. Qi, J. Nonlinear Sci. Appl. 9 (2016), 1872–1876
1874
Proof. From Theorem 2.1, it follows that
n n X
X
d
n
n−1
Chn (x|λ) =
Chm (λ)(n − m)xn−m−1 = n
Chm (λ)xn−m−1
dx
m
m−1
m=1
m=1
n−1
X n − 1
=n
Chn−m−1 (λ)xm = nChn−1 (x|λ).
m
m=0
This means that Chn (x|λ) is an Appel sequence. Furthermore, we observe that
Z
y ln(1+λt)/λ+xt
e
Zp
∞
X
! ∞
!
X xl
1
m
l
d µ−1 (y) =
y µ−1 (y) (ln(1 + λt))
t
λ
m!
l!
Zp
m=0
l=0
! ∞
!
∞
∞
k tk
X xl
X
X
λ
=
tl
λ−m Em (0)
S1 (k, m)
k!
l!
m=0
l=0
k=m
! ! ∞
!
∞
k
k
X xl
X X
t
=
λk−m Em (0)S1 (k, m)
tl
k!
l!
l=0
k=0 m=0
!
∞
n
k
n
X XX n
k−m
n−k t
.
=
λ
Em (0)S1 (k, m)x
n!
k
n=0
−m
Z
m
k=0 m=0
Combining this with (2.1) yields the required identity.
Theorem 2.3. For n ≥ 0, we have
n
X
Chm (x|λ)λn−m S2 (n, m) =
m=0
n X
n
m=0
m
Bm
x m
λ En−m (0),
λ
where S2 (m, n) is the Stirling number of the second kind.
Proof. By replacing t by
eλt −1
λ
in (2.1), we obtain
∞
X
2
1 eλt − 1 m
x(eλt −1)/λ
e
=
Chm (x|λ)
et + 1
m!
λ
m=0
∞
X
∞
1 X
λn tn
S
(n,
m)
2
λm n=m
n!
m=0
"
#
∞ X
n
X
tn
n−m
=
.
Chm (x|λ)λ
S2 (n, m)
n!
=
Chm (x|λ)
(2.2)
n=0 m=0
Recall from [1, p. 265] that the Bell polynomials Bn (x) are generated by
e
x(et −1)
=
∞
X
Bn (x)
n=0
tn
.
n!
Therefore, we acquire that
2
λt
ex(e −1)/λ =
t
e +1
∞
X
El (0)
l=0
l!
!
t
l
∞
X
m=0
!
m m! X
∞
n X
x λ t
n
x m
tn
Bm
=
Bm
λ En−m (0)
.
λ
m!
m
λ
n!
Comparing this with (2.2) leads to the required identity.
n=0
m=0
D. Lim, F. Qi, J. Nonlinear Sci. Appl. 9 (2016), 1872–1876
1875
(r)
For r ∈ N, define the higher order λ-Changhee polynomials Chn (x) by
Z
Z
···
e
(x1 +···+xr ) ln(1+λt)/λ+xt
Zrp
2
d µ−1 (x1 ) · · · d µ−1 (xr ) =
(1 + λt)1/λ + 1
(r)
r
e
xt
=
∞
X
Ch(r)
n (x|λ)
n=0
(r)
When x = 0, we call Chn (λ) = Chn (0|λ) the higher order λ-Changhee numbers.
Theorem 2.4. For n ≥ 1, we have
Ch(r)
n (x|λ)
=
n X
n
m
m=0
n−m
Ch(r)
m (λ)x
d
(r)
Chn(r) (x|λ) = nChn−1 (x|λ).
dx
and
Proof. This follows from the observation that
∞
X
tn
Ch(r)
n (x|λ)
2
=
n!
(1 + λt)1/λ + 1
n=0
r
xt
e
" n #
∞ X
n
X
n
(r)
n−m t
=
Chm (λ)x
.
m
n!
n=0 m=0
(r)
Recall from [8, p. 12] that the higher order Euler polynomials En (x) may be represented by
Z
Z
(x1 +···+xr +x)t
···
e
Zrp
2
d µ−1 (x1 ) · · · d µ−1 (xr ) t
e +1
r
ext =
∞
X
En(r) (x)
n=0
tn
.
n!
(r)
When x = 0, we call En (0) the higher order modified Euler numbers.
Theorem 2.5. For n ≥ 0, we have
Ch(r)
n (x|λ)
n X
k X
n k−m (r)
=
λ
Em (0)S1 (k, m)xn−k .
k
k=0 m=0
Proof. We observe that
Z
Z
···
e(x1 +···+xr ) ln(1+λt)/λ+xt d µ−1 (x1 ) · · · d µ−1 (xr )
Zrp
∞ Z
X
[ln(1 + λt)]m
=
···
(x1 +· · ·+ xr ) d µ−1 (x1 ) · · · d µ−1 (xr )
m!λm
Zrp
m=0
!
!
∞
∞
∞
X
X
X
xl l
λk tk
(r)
=
t
λ−m Em
(0)
S1 (k, m)
k!
l!
m=0
l=0
k=m
!
! ! ∞
∞
k
k
X xl
X
X
t
(r)
tl
=
λk−m Em
(0)S1 (k, m)
k!
l!
k=0 m=0
l=0
!
∞
n
k
n
X XX n
k−m (r)
n−k t
=
λ
Em (0)S1 (k, m)x
.
k
n!
n=0
Z
m
k=0 m=0
Combination of this identity with (2.3) results in the required identity.
Theorem 2.6. For n ≥ 0, we have
n
X
m=0
n−m
λ
Ch(r)
m (x|λ)S2 (n, m)
=
n
X
m=0
x m (r)
Bm
λ En−m (0).
λ
!
∞
X
xl
l=0
l!
tl
tn
. (2.3)
n!
D. Lim, F. Qi, J. Nonlinear Sci. Appl. 9 (2016), 1872–1876
Proof. Substituting
2
t
e +1
r
eλt −1
λ
for t in (2.3) gives
λt −1)/λ
ex(e
∞
X
=
Ch(r)
m (x|λ)
m=0
=
∞
X
m=0
1876
1
Ch(r)
m (x|λ) m
λ
m
1 1 λt
e −1
m
λ m!
∞
X
" n
#
∞
λn tn X X n−m (r)
tn
S2 (n, m)
=
λ
Chm (x|λ)S2 (n, m)
.
n!
n!
n=m
n=0 m=0
On the other hand,
2
t
e +1
r
x(eλt −1)/λ
e
=
"∞
X
l=0
tl
(r)
El (0)
l!
#"
∞
X
m=0
" n
#
m m# X
∞ X
x λ t
x m (r)
tn
Bm
=
Bm
λ En−m (0)
.
λ
m!
λ
n!
n=0 m=0
The required result thus follows.
Remark 2.7. This paper is a slightly modified version of the preprint [7].
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