Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 865721, 12 pages
doi:10.1155/2012/865721
Research Article
Applications of Umbral Calculus Associated with
p-Adic Invariant Integrals on Zp
Dae San Kim1 and Taekyun Kim2
1
2
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr
Received 8 November 2012; Accepted 22 November 2012
Academic Editor: Gaston N’Guerekata
Copyright q 2012 D. S. Kim and T. Kim. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Recently, Dere and Simsek 2012 have studied the applications of umbral algebra to some special
functions. In this paper, we investigate some properties of umbral calculus associated with p-adic
invariant integrals on Zp . From our properties, we can also derive some interesting identities of
Bernoulli polynomials.
1. Introduction
Let p be a fixed prime number. Throughout this paper, Zp , Qp , and Cp denote the ring of padic integers, the field of p-adic rational numbers, and the completion of algebraic closure of
Qp , respectively.
Let N ∪ {0}. Let UDZp be space of uniformly differentiable functions on Zp . For
f ∈ UDZp , the p-adic invariant integral on Zp is defined by
p −1
1 fxdμx lim N
fx,
N →∞p
Zp
x0
N
1.1
see 1, 2.
From 1.1, we have
Zp
fx ndμx −
Zp
fxdμx n
l0
f l,
n ∈ N,
1.2
2
Abstract and Applied Analysis
where f l dfx/dx|xl see 1–6. Let F be the set of all formal power series in the
variable t over Cp with
F
ft ∞
ak
k0
k!
t | ak ∈ Cp .
k
1.3
Let P Cp x and let P∗ denote the vector space of all linear functional on P.
The formal power series,
ft ∞
ak
k!
k0
tk ∈ F,
1.4
defines a linear functional on P by setting
ft | xn an ,
∀n ≥ 0,
1.5
see 7, 8.
In particular, by 1.4 and 1.5, we get
tk | xn n!δn,k ,
1.6
where δn,k is the Kronecker symbol see 7. Here, F denotes both the algebra of formal
power series in t and the vector space of all linear functional on P, so an element ft of F will
be thought of as both a formal power series and a linear functional. We shall call F the umbral
algebra. The umbral calculus is the study of umbral algebra.
The order oft of power series ft /
0 is the smallest integer k for which ak does
not vanish. We define oft ∞ if ft 0. From the definition of order, we note that
oftgt oft ogt and oft gt ≥ min{oft, ogt}.
The series ft has a multiplicative inverse, denoted by ft−1 or 1/ft, if and only if
oft 0.
Such a series is called invertible series. A series ft for which oft 1 is called a
delta series see 7, 8. Let ft, gt ∈ F. Then, we have
ftgt | px ft | gtpx gt | ftpx .
1.7
By 1.5 and 1.6, we get
eyt | xn yn ,
eyt | px p y ,
1.8
see 7.
Notice that for all ft in F,
∞ ft | xk
ft tk ,
k!
k0
1.9
Abstract and Applied Analysis
3
and for all polynomials px,
tk | px
xk ,
px k!
k≥0
1.10
see 7, 8.
Let f1 t, f2 t, . . . , fm t ∈ F. Then, we have
f1 tf2 t · · · fm t | xn n
f1 t | xi1 · · · fm t | xim ,
i 1 , . . . , im
1.11
where the sum is over all nonnegative integers i1 , i2 , . . . , im such that i1 · · · im n see 8.
By 1.10, we get
pk x n tl | px
dk px ll − 1 · · · l − k 1xl−k .
l!
dxk
lk
1.12
Thus, from 1.12, we have
pk 0 tk | px 1 | pk x ,
1.13
see 7.
By 1.13, we get
t px p
k
k
x dk px
dxk
.
1.14
Thus, by 1.14, we see that
eyt px p x y .
1.15
Let us assume that sn x is a polynomial of degree n. Suppose that ft, gt ∈ F with
oft 1 and ogt 0. Then, there exists a unique sequence sn x of polynomials
satisfying gtftk | sn x n!δn,k for all n, k ≥ 0.
The sequence sn x is called the Sheffer sequence for gt, ft, which is denoted by
sn x ∼ gt, ft.
The Sheffer sequence for gt, t is called the Appell sequence for gt, or sn x is
Appell for gt, which is indicated by sn x ∼ gt, t.
4
Abstract and Applied Analysis
For px ∈ P, it is known that
ft | xpx ∂t ft | px f t | px ,
yt
e − 1 | px p y − p0,
1.16
see 7, 8.
Let sn x ∼ gt, ft. Then, we have
ht ∞
ht | sk x
k!
k0
px ∞
gtftk | px
k!
k0
1
gtftk ,
e
g ft
yft
∞ s y
k
tk ,
k!
k0
ht ∈ F,
1.17
px ∈ P,
1.18
for any y ∈ Cp ,
1.19
sk x,
where ft is the compositional inverse of ft, and
ftsn x nsn−1 x,
1.20
see 7, 8.
We recall that the Bernoulli polynomials are defined by the generating function to be
∞
tn
t
xt
Bxt
e
e
Bn x ,
t
n!
e −1
n0
1.21
with the usual convention about replacing Bn x by Bn x see 1–16.
In the special case, x 0, Bn 0 Bn are called the nth Bernoulli numbers. By 1.21,
we easily get
Bn x B xn n n
l
l0
Bl xn−l n n
l0
l
Bn−l xl .
1.22
Thus, by 1.22, we see that Bn x is a monic polynomial of degree n. It is easy to show that
B0 1,
Bn 1 − Bn δ1,n ,
1.23
see 13–15.
From 1.2, we can derive the following equation:
Zp
fx 1dμx −
Zp
fxdμx f 0.
1.24
Abstract and Applied Analysis
5
Let us take fx etx ∈ UDZp . Then, from 1.21, 1.22, 1.23, and 1.24, we have
x dμx Bn ,
Zp
n x y dμ y Bn x,
n
Zp
1.25
where n ≥ 0 see 1, 2. Recently, Dere and simsek have studied applications of umbral
algebra to some special functions see 7. In this paper, we investigate some properties of
umbral calculus associated with p-adic invariant integrals on Zp . From our properties, we can
derive some interesting identities of Bernoulli polynomials.
2. Applications of Umbral Calculus Associated with p-Adic
Invariant Integrals on Zp
Let sn x be an Appell sequence for gt. By 1.19, we get
1 n
x sn x,
gt
iff xn gtsn x.
2.1
Let us take gt et − 1/t ∈ F. Then, gt is clearly invertible series. From 1.21 and 2.1,
we have
∞
Bk x
k!
k0
tk 1 xt
e .
gt
2.2
Thus, by 2.2, we get
1 n
x Bn x,
gt
tBn x Bn x nBn−1 x,
n ≥ 0.
2.3
From 1.21, 2.1, and 2.3, we note that Bn x is an Appell sequence for gt et − 1/t.
Let us take the derivative with respect to t on both sides of 2.2. Then, we have
∞
Bk x
k1
k!
ktk−1 xgtext − ext g t
gt2
∞
k0
xk
xk g t
x
−
gt gt gt
tk
.
k!
2.4
Thus, by 2.4, we get
xk g t
xk
Bk1 x x
−
gt gt gt
g t
x−
gt
Bk x,
2.5
6
Abstract and Applied Analysis
where k ≥ 0.
Zp
e
xy1t
dμ y −
Zp
exyt dμ y text .
2.6
Thus, by 2.6, we get
Zp
n x y 1 dμ y −
n x y μ y nxn−1 ,
Zp
n ≥ 0.
2.7
From 1.25 and 2.7, we have
Bn x 1 − Bn x nxn−1 ,
2.8
n ≥ 0.
By 2.5, we see that
gtBk1 x gtxBk x − g tBk x,
2.9
Thus, by 2.9, we have
et − 1 Bk1 x et − 1 xBk x − et − gt Bk x,
k ≥ 0,
2.10
and we can derive the following equation.
From 2.3 and 2.10,
Bk1 x 1 − Bk1 x x 1Bk x 1 − xBk x − Bk x 1 xk ,
k ≥ 0.
2.11
By 2.8 and 2.11, we see that
Bk1 x 1 Bk1 x k 1xk .
2.12
Therefore, by 2.5, we obtain the following theorem.
Theorem 2.1. For k ∈ Z , one has
Bk1 x x−
g t
Bk ,
gt
2.13
where g t dgt/dt.
Corollary 2.2. For ≥ 0, one has
Bk1 x 1 Bk1 x k 1xk .
2.14
Abstract and Applied Analysis
7
Let us consider the linear functional ft that satisfies
ft | px pudμu,
Zp
2.15
for all polynomials px. It can be determined from 1.9 that
∞ ft | xk
∞ tk
k
ft t uk dμu
k!
k!
k0
k0 Zp
eut dμu.
2.16
Zp
By 1.24 and 2.16, we get
ft Zp
eut dμu et
t
.
−1
2.17
Therefore, by 2.17, we obtain the following theorem.
Theorem 2.3. For px ∈ P, one has
Zp
e dμu | px
ut
Zp
pudμu.
2.18
That is
t
| px
et − 1
Zp
pudμu.
2.19
In particular, one has
Bn e dμu | x
ut
Zp
n
.
2.20
From 1.24, one has
n0
n tn
x y dμ y
n!
Zp
Zp
exyt dμ y
∞ n0
tn
e dμ y xn .
n!
Zp
yt
2.21
8
Abstract and Applied Analysis
By 1.25 and 2.21, we get
Bn x n x y dμ y Zp
Zp
eyt dμ y xn ,
2.22
where n ≥ 0.
Therefore, by 2.22, we obtain the following theorem.
Theorem 2.4. For px ∈ P, we have
Zp
p x y dμ y Zp
eyt dμ y px
2.23
t
px.
t
e −1
In particular, one obtains
Bn x x y dμ y Zp
n
Zp
eyt dμ y xn
2.24
t
xn .
t
e −1
r
The higher order Bernoulli polynomials Bn x are defined by
···
Zp
Zp
ex1 x2 ···xr xt dμx1 · · · dμxr t
t
e −1
r
ext
∞
tn
r
Bn x .
n!
n0
r
2.25
r
In the special case, x 0, Bn 0 Bn are called the nth Bernoulli numbers of order r ∈ N.
From 2.25, we note that
Zp
···
Zp
x1 · · · xr n dμx1 · · · dμxr i1 ···ir n
i1 ···ir n
n
i 1 , . . . , ir
Zp
x1i1 dμx1 Zp
n
r
B · · · Bir Bn .
i1 , . . . , ir i1
x2i2 dμx2 · · ·
Zp
xrir dμxr 2.26
Abstract and Applied Analysis
9
By 2.25 and 2.26, we get
r
Bn x
n n
l
l0
r
Bn−l xl .
2.27
r
From 2.26 and 2.27, we note that Bn x is a monic polynomial of degree n with
coefficients in Q. For r ∈ N, let us assume that
g
r
t −1
Zp
···
e
Zp
x1 ···xr t
dμx1 · · · dμxr et − 1
t
r
.
2.28
By 2.28, we easily see that g r t is an invertible series. From 2.25 and 2.28, we have
ext
g r t
r
Zp
···
Zp
ex1 ···xr xt dμx1 · · · dμxr ∞
tn
r
Bn x ,
n!
n0
2.29
r
tBn x nBn−1 x.
r
From 2.29, we note that Bn is an Appell sequence for g r t. Therefore, by 2.29, we obtain
the following theorem.
Theorem 2.5. For px ∈ P and r ∈ N, one has
Zp
···
Zp
px1 · · · xr xdμx1 · · · dμxr t
et − 1
r
px.
2.30
ex1 ···xr t dμx1 · · · dμxr xn .
2.31
In particular, the Bernoulli polynomials of order r are given by
r
Bn x t
et − 1
r
xn Zp
···
Zp
That is
r
Bn x
∼
et − 1
t
r ,t .
2.32
Let us consider the linear functional f r t that satisfies
f r t | px Zp
···
Zp
px1 · · · xr dμx1 · · · dμxr ,
2.33
10
Abstract and Applied Analysis
for all polynomials px. It can be determined from 1.9 that
∞ f r
t | xk k
r
t
f t k!
k0
∞ Zp
k0
tk
k!
2.34
ex1 ···xr t dμx1 · · · dμxr Zp
t
t
e −1
x1 · · · xr k dμx1 · · · dμxr Zp
···
Zp
···
r
.
Therefore, by 2.34, we obtain the following theorem.
Theorem 2.6. For px ∈ P, one has
Zp
···
That is
t
et − 1
r
e
Zp
x1 ···xr t
dμx1 · · · dμxr | px
2.35
Zp
···
Zp
| px px1 · · · xr dμx1 · · · dμxr .
Zp
···
Zp
px1 · · · xr dμx1 · · · dμxr .
2.36
In particular, one gets
r
Bn
Zp
···
Zp
e
x1 ···xr t
dμx1 · · · dμxr | x
n
2.37
.
Remark 2.7. From 1.11, we note that
x1 ···xr t
n
···
e
dμx1 · · · dμxr | x
Zp
Zp
ni1 ···ir
n
i 1 , . . . , ir
Zp
e
x1 t
dμx1 | x
i1
···
2.38
e dμxr | x
xr t
Zp
ir
.
By Theorems 2.3 and 2.6 and 2.38, we get
r
Bn ni1 ···ir
n
B · · · Bir .
i1 , . . . , ir i1
Let sn x be the Sheffer sequence for gt, ft.
2.39
Abstract and Applied Analysis
11
Then the Sheffer identity is given by
n n
sn x y pk y sn−k x,
k
k0
2.40
see 7, 8, where pk y gtsk y. From Theorem 2.5 and 2.40, we have
n n r
xy Bn−k xxk .
k
k0
r Bn
2.41
Acknowledgments
This research was supported by Basic Science Research Program through the National
Research Foundation of Korea NRF funded by the Ministry of Education, Science and
Technology 2012R1A1A2003786.
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