R E S E A R C H
Open Access
Some identities of Korobov-type
polynomials associated with
p-adic integrals
on
Z
p
Dae San Kim
1and Taekyun Kim
2**Correspondence: [email protected] 2Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea Full list of author information is available at the end of the article
Abstract
In this paper, we consider Korobov-type polynomials derived from the bosonic and fermionicp-adic integrals onZp, and we give some interesting and new identities of
those polynomials and of their mixed-types.
MSC: 11B68; 11B83; 11S80; 05A19
Keywords: Korobov-type polynomial;p-adic integral;
λ-Changhee and Korobov
mixed-type polynomial; Korobov andλ-Changhee mixed-type polynomial
1 Introduction
Letpbe a fixed odd prime number. Throughout this paper,Zp,Qp andCpwill denote
the ring ofp-adic integers, the field ofp-adic rational numbers and the completion of
the algebraic closure ofQp. Letνp be the normalized exponential valuation ofCp with |p|p=p–νp(p)=p. LetUD(Zp) be the space of uniformly differentiable functions onZp. For f∈UD(Zp), the bosonicp-adic integral onZpis defined by
I(f) =
Zp
f(x)dμ(x)
= lim
N→∞ pN–
x=
f(x)μ
x+pNZp
= lim
N→∞
pN pN–
x=
f(x). (.)
Thus, by (.), we get
I(f) =I(f) +f(), wheref(x) =f(x+ ) (see [–]). (.)
The fermionicp-adic integral onZpis defined by Kim as
I–(f) =
Zp
f(x)dμ–(x)
= lim
N→∞
pN–
x=
f(x)μ–
x+pNZp
= lim
N→∞ pN–
x=
f(x)(–)x. (.)
Thus, from (.), we have
I–(f) = –I–(f) + f() (see []). (.)
From (.) and (.), we can derive the following equations:
I(fn) –I(f) =
n–
l=
f(l), I–(fn) + (–)n–I–(f) =
n–
l=
f(l), (.)
wherefn(x) =f(x+n),f(l) =dfdx(x)|x=l(see [, , ]).
As is well known, the Bernoulli polynomials of orderr(∈N) are defined by the generat-ing function
t et–
r ext=
∞
n=
B(nr)(x)t
n
n! (see [, ]). (.)
Whenx= ,B(nr)=B(nr)() are called the Bernoulli numbers of orderr. In particular, if r= ,Bn(x) =B()n (x) are called the ordinary Bernoulli polynomials.
The Euler polynomials of orderrare also given by the generating function
et+
r ext=
∞
n=
En(r)(x)t
n
n! (see [, ]). (.)
Whenx= ,E(nr)=En(r)() are called the Euler numbers of orderr. In particular, ifr= ,
thenEn(x) =E()n (x) are called the ordinary Euler polynomials.
The Daehee polynomials of orderrare defined by the generating function
log ( +t)
t
r
( +t)x= ∞
n=
D(nr)(x)t
n
n! (see []). (.)
Whenx= ,D(nr)=D(nr)() are called the Daehee numbers of orderr. In particular, ifr= ,
thenDn(x) =D()n (x) are called the ordinary Daehee polynomials. Now, we introduce the
Changhee polynomials of orderrgiven by the generating function
t+ r
( +t)x= ∞
n=
Ch(nr)(x)t
n
Whenx= ,Ch(nr)=Ch(nr)() are called the Changhee numbers of orderr. In particular, ifr= , thenChn(x) =Ch()n (x) are called the ordinary Changhee polynomials.
Recently, Korobov introduced the special polynomials given by the generating function
λt
(t+ )λ– ( +t) x=
∞
n=
Kn(x|λ) tn
n! (λ∈N) (see [–]). (.)
Note thatlimλ→Kn(x|λ) =bn(x), wherebn(x) are the Bernoulli polynomials of the
sec-ond kind defined by the generating function
t
log( +t)
( +t)x= ∞
n=
bn(x) tn
n! (see [, ]). (.)
In this paper, we define the higher-order Korobov polynomials given by the generating function
λt
(t+ )λ–
r
( +t)x= ∞
n=
Kn(r)(x|λ)t
n
n!. (.)
Whenx= ,Kn(r)(λ) =Kn(r)(|λ) are called the Korobov numbers of orderr. In particular,
ifr= , thenKn(λ) =Kn()(|λ) =Kn(|λ) are called the ordinary Korobov numbers. Now,
we consider the Korobov-type Changhee polynomials which are called theλ-Changhee polynomials as follows:
( +t)λ+
( +t)x= ∞
n=
Chn(x|λ) tn
n!. (.)
When x = , Chn(λ) = Chn( | λ) are called λ-Changhee numbers. Note that
limλ→Chn(x|λ) =Chn(x),limλ→Chn(x|λ) = (x)n, where
(x)n=x(x– )· · ·(x–n+ ) = n
l=
S(n,l)xl (see []).
Forr∈N, theλ-Changhee polynomials of orderrare defined by the generating function
( +t)λ+
r
( +t)x= ∞
n=
Ch(nr)(x|λ)t
n
n!. (.)
The Stirling numbers of the second kind are defined by the generating function
et– n=n! ∞
l=n S(l,n)
tl
l! (see [, ]). (.)
Bernoulli polynomials, Euler polynomials and their variants by means of generating func-tions,p-adic integrals and umbral calculus (see [, , , ]).
Here in this paper we introduce two Korobov-type polynomials obtained from the same function, namely the one by performing bosonic p-adic integrals on Zp and the other
by carrying out fermionicp-adic integrals onZp. In addition, we consider their
higher-order versions and some mixed-types of them by considering multivariate p-adic inte-grals. In conclusion, we will obtain some connections between these new polynomials and Bernoulli polynomials, Euler polynomials, Daehee numbers and Bernoulli numbers of the second kind.
2 Korobov-type polynomials
Forλ∈N, by (.), we get
Zp
( +t)λy+xdμ(y) =
λlog( +t) ( +t)λ– ( +t)
x
=
λt
( +t)λ–
log( +t)
t
( +t)x
=
∞
l=
Kl(x|λ) tl l!
∞
m=
Dm tm m!
= ∞
n=
n
l=
Kl(x|λ)Dn–l n!
l!(n–l)!
tn n!
= ∞
n=
n
l=
n l
Kl(x|λ)Dn–l tn
n!. (.)
From (.), we have
Zp
λy+x n
dμ(y) =
n!
n
l=
n l
Kl(x|λ)Dn–l. (.)
Therefore, by (.), we obtain the following theorem.
Theorem . For n≥,we have
Zp
λy+x n
dμ(y) =
n!
n
l=
n l
Kl(x|λ)Dn–l.
Now, we observe that
Zp
(λy+x)ndμ(y) =
n
l=
S(n,l)
Zp
(λy+x)ldμ(y)
=
n
l=
S(n,l)λl
Zp
y+x
λ
l dμ(y)
=
n
l=
S(n,l)λlBl
x λ
Therefore by (.), we obtain the following corollary.
Therefore, by (.), we obtain the following corollary.
Corollary . For n≥,we have
On the other hand,
∞
Theorem . For n≥,we have
n
l=
n l
λn–lBn–l
x λ
l+ =
n
m=
Km(x|λ)S(n,m).
It is easy to show that
Zp
f(x)dμ(x) =
d d–
a=
Zp
f(a+dx)dμ(x) (d∈N). (.)
By (.), we get
Zp
( +t)λxdμ(x) =
d d–
a=
Zp
( +t)(a+dx)λdμ
(x)
=
d d–
a=
( +t)aλ
Zp
( +t)λdxdμ(x)
=
d d–
a=
( +t)aλλdlog( +t)
( +t)dλ– . (.)
On the other hand,
Zp
( +t)λxdμ(x) =
λlog( +t)
( +t)λ– . (.)
Thus, by (.) and (.), we get
λt
( +t)λ– ( +t) x=
d d–
a=
λdt
( +t)λd– ( +t)
aλ+x. (.)
Therefore, by (.) and (.), we obtain the following theorem.
Theorem . For n≥and d∈N,we have
Kn(x|λ) =
d d–
a=
Kn(aλ+x|λd).
From (.), we can derive the following equation:
Zp
( +t)(x+n)λdμ(x) –
Zp
( +t)λxdμ(x) =λlog( +t)
n–
l=
( +t)λl (n∈N). (.)
Thus, by (.), we get
Zp
( +t)λxdμ(x) =
λlog( +t) ( +t)λn–
n–
l=
From (.), we have
On the other hand,
t
Therefore, by (.) and (.), we obtain the following theorem.
Theorem . For n∈N,m≥,we have
Remark By (.), we easily get
Now, we consider the multivariatep-adic integral onZpgiven by
Thus, by (.), we get
By comparing the coefficients on both sides, we obtain the following theorem.
Theorem . For n≥,we have
On the other hand,
Therefore, by (.) and (.), we obtain the following theorem.
From (.), we can derive the following equation:
We observe that
∞
Therefore, by (.) and (.), we obtain the following theorem.
On the other hand,
Therefore, by (.), (.), and (.), we obtain the following theorem.
Theorem . For n≥,we have
From (.), we can derive the following equation:
∞
Therefore, by (.) and (.), we obtain the following theorem.
Let us observe the following multivariate fermionicp-adic integral onZp:
Zp
· · ·
Zp
( +t)λ(x+···+xr)+xdμ
–(x)· · ·dμ–(xr)
=
( +t)λ+
r
( +t)x
= ∞
n=
Ch(nr)(x|λ)t
n
n!. (.)
Thus, by (.), we get
Ch(nr)(x|λ)
n!
=
Zp
· · ·
Zp
λ(x+· · ·+xr) +x n
dμ–(x)· · ·dμ–(xr) (n≥). (.)
Note that
∞
n=
Ch(nr)(x|λ)t
n n! =
( +t)λ+
r
( +t)x= ∞
n=
n
m=
(x)mCh(nr–)m(λ)
n m
tn n!.
Thus, we get
Ch(nr)(x|λ) =
n
m=
(x)mCh(nr–)m(λ)
n m
(n≥). (.)
By (.) and (.), we easily get
Ch(nr)(x|λ) =
Zp
· · ·
Zp
(λx+· · ·+λxr+x)ndμ–(x)· · ·dμ–(xr)
=
n
l=
S(n,l)λl
Zp
· · ·
Zp
x+· · ·+xr+ x λ
l
dμ–(x)· · ·dμ–(xr)
=
n
l=
S(n,l)λlE(lr)
x λ
. (.)
Now, we consider the λ-Changhee and Korobov mixed-type polynomials which are given by the multivariatep-adic integral onZpas follows:
CK(nr,s)(x|λ)
=
Zp
· · ·
Zp
Ch(nr)(λx+· · ·+λxs+x|λ)dμ(x)· · ·dμ(xs)
=
n
m=
Ch(nr–)m(λ)
n m Zp
· · ·
Zp
(λx+· · ·+λxs+x)mdμ(x)· · ·dμ(xs), (.)
Now, we observe that
Zp
· · ·
Zp
( +t)λx+···+λxs+xdμ
(x)· · ·dμ(xs)
=
λt
( +t)λ–
slog
( +t)
t
s
( +t)x
= ∞
n=
n
l=
Kl(s)(x|λ)D(ns–)l
n l
tn
n!. (.)
By (.), we get
Zp
· · ·
Zp
(λx+· · ·+λxs+x)ndμ(x)· · ·dμ(xs)
=
n
l=
Kl(s)(x|λ)D(ns–)l
n l
. (.)
From (.) and (.), we have
CK(nr,s)(x|λ) =
n
m=
m
l=
m l
n m
Ch(nr–)m(λ)Kl(s)(x|λ)D(ms)–l. (.)
The generating function ofCK(nr,s)(x|λ) is given by
∞
n=
CK(nr,s)(x|λ)t
n n!
= ∞
n=
Zp
· · ·
Zp
Ch(nr)(λx+· · ·+λxs+x|λ)dμ(x)· · ·dμ(xs) tn n!
=
Zp
· · ·
Zp
r-times
Zp
· · ·
Zp
s-times
( +t)λy+···+λyr+λx+···+λxs+x
×dμ–(y)· · ·dμ–(yr)dμ(x)· · ·dμ(xs)
=
( +t)λ+
rλlog
( +t) ( +t)λ–
s
( +t)x. (.)
Theorem . For r,s∈Nand n≥,we have
CK(nr,s)(x|λ) =
n
m=
m
l=
m l
n m
Ch(nr–)m(λ)Kl(s)(x|λ)D(ms)–l.
We consider the Korobov andλ-Changhee mixed-type polynomials, which are given by
KC(nr,s)(x|λ) =
Zp
· · ·
Zp
Kn(r)(λx+· · ·+λxs+x|λ)dμ–(x)· · ·dμ–(xs), (.)
Then, by (.), we get
KC(nr,s)(x|λ)
=
n
m=
n m
Km(r)(λ)
Zp
· · ·
Zp
(λx+· · ·+λxs+x)n–mdμ–(x)· · ·dμ–(xs)
=
n
m=
n m
Km(r)(λ)Chn(s–)m(x|λ). (.)
The generating function ofKC(nr,s)(x|λ) is given by
∞
n=
KC(nr,s)(x|λ)t
n n!
= ∞
n=
Zp
· · ·
Zp
Kn(r)(λx+· · ·+λxs+x|λ)dμ–(x)· · ·dμ–(xs) tn n!
=
λt
( +t)λ–
r
Zp
· · ·
Zp
( +t)λx+···+λxs+xdμ
–(x)· · ·dμ–(xs)
=
λt
( +t)λ–
r
( +t)λ+
s
( +t)x. (.)
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea.2Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea.
Acknowledgements
The authors would like to thank the referees and editor for their valuable comments. The work reported in this paper was conducted during the sabbatical year of Kwangwoon University in 2014.
Received: 1 February 2015 Accepted: 10 August 2015
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