Applications on the Apostol Daehee numbers and polynomials associated with special numbers, polynomials, and p adic integrals Simsek and Yardimci Advances in Difference Equations (2016) 2016 308 DOI 1[.]
Trang 1R E S E A R C H Open Access
Applications on the Apostol-Daehee
numbers and polynomials associated with
special numbers, polynomials, and p-adic
integrals
Yilmaz Simsek1* and Ahmet Yardimci1,2
* Correspondence:
ysimsek@akdeniz.edu.tr
1 Department of Mathematics,
Faculty of Science, University of
Akdeniz, Antalya, 07058, Turkey
Full list of author information is
available at the end of the article
Abstract
In this paper, by using p-adic Volkenborn integral, and generating functions, we give
some properties of the Bernstein basis functions, the Apostol-Daehee numbers and polynomials, Apostol-Bernoulli polynomials, some special numbers including the Stirling numbers, the Euler numbers, the Daehee numbers, and the Changhee numbers By using an integral equation and functional equations of the generating functions and their partial differential equations (PDEs), we give a recurrence relation for the Apostol-Daehee polynomials We also give some identities, relations, and integral representations for these numbers and polynomials By using these relations,
we compute these numbers and polynomials We make further remarks and observations for special polynomials and numbers, which are used to study elementary word problems in engineering and in medicine
MSC: 12D10; 11B68; 11S40; 11S80; 26C05; 26C10; 30B40; 30C15 Keywords: Bernoulli numbers and polynomials; Apostol-Bernoulli numbers and
polynomials; Daehee numbers and polynomials; Apostol-Daehee numbers; array polynomials; Stirling numbers of the first kind and the second kind; generating function; functional equation; derivative equation; Bernstein basis functions
1 Introduction
The special numbers and polynomials have been used in various applications in such di-verse areas as mathematics, probability and statistics, mathematical physics, and engineer-ing For example, due to the relative freedom of some basic operations including addition, subtraction, multiplication, polynomials can be seen almost ubiquitously in engineering They are curves that represent properties or behavior of many engineering objects or de-vices For example, polynomials are used in elementary word problems to complicated problems in the sciences, approximate or curve fit experimental data, calculate beam de-flection under loading, represent some properties of gases, and perform computer aided geometric design in engineering Polynomials are used as solutions of differential equa-tions Polynomials represent characteristics of linear dynamic system and we also know that a ratio of two polynomials represents a transfer function of a linear dynamic system
© The Author(s) 2016 This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
Trang 2With the help of polynomials, one defines basis functions used in finite element
compu-tation and constructs parametric curves
In order to give some results including identities, relations, and formulas for special
numbers and polynomials, we use the p-adic Volkenborn integral and generating
func-tion methods We need the following formulas, relafunc-tions, generating funcfunc-tions, and
nota-tions for families of special numbers and polynomials Throughout this paper, we use the
following notations:
LetC, R, Z, and N be the sets of complex numbers, real numbers, integers, and positive integers, respectively, andN=N ∪ {} and Z–={–, –, –, } Also let Zp be the set
of p-adic integers We assume that ln(z) denotes the principal branch of the multi-valued
function ln(z) with the imaginary part Im(ln(z)) constrained by –π < Im(ln(z)) ≤ π
Fur-thermore, n = if n = , and n = if n∈ N We have
x v
=x (x – ) · · · (x – v + )
(x) v
v! (x ∈ C; v ∈ N)
(cf [–], and the references cited therein).
There are many methods and techniques for investigating and constructing generating functions for special polynomials and numbers One of the most important techniques is
the p-adic Volkenborn integral onZp In [], Kim constructed the p-adic q-Volkenborn
integration By using this integral, we derive some identities, and relations for the special
polynomials We now briefly give some definitions and properties of this integral
Let f ∈ UD(Z p), the set of uniformly differentiable functions on Zp The p-adic q-Volkenborn integration of f onZpis defined by Kim [] as follows:
Zp
f (x) dμ q (x) = lim
N→∞
[p N]q
pN–
x=
where
[x] =
–q x
–q, q= ;
x, q=
and μ q (x) denotes the q-distribution onZp, which is given by
μ q
x + p NZp
x
[p N]q
,
where q∈ Cpwith| – q| p < (cf []).
If q → in (.), then we have the bosonic p-adic integral (p-adic Volkenborn integral), which is given by (cf [, ])
Zp
f (x) dμ(x) = lim
N→∞
p N
pN–
x=
where
μ
x + p NZp
p N
Trang 3By using this integral, the Bernoulli polynomials are given by
B n (x) =
Zp
(cf [, , , , ], and the references cited therein).
The Bernoulli polynomials B n (x) are also defined by means of the following generating
function:
F B (t, x) = t
e t– e
tx=
∞
n=
B n (x) t
n
n!
with B n () = B n , which denotes the Bernoulli numbers (of the first kind) (cf [, , –,
, , , ], and the references cited therein)
If q → – in (.), then we have the fermionic p-adic integral on Z p given by (cf []):
Zp
f (x) dμ–(x) = lim
N→∞
pN–
x=
where p= and
μ–
x + p NZp
= (–)x
(cf []) By using (.), we have the Witt formula for the Euler numbers E nas follows:
E n (x) =
Zp
(cf [, , , ], and the references cited therein).
The Euler polynomials E n (x) are also defined by means of the following generating
func-tion:
F E (t, x) =
e t+ e
tx=
∞
n=
E n (x) t
n
n!
with E n () = E n , which denotes the Euler numbers (of the first kind) (cf [, , –, ,
, , ], and the references cited therein)
The λ-Bernoulli numbers and polynomials have been studied in different sets For in-stance on the set of complex numbers, we assume that λ ∈ C and on set of p-adic numbers
or p-adic integrals, we assume that λ∈ Zp
The Apostol-Bernoulli polynomials B n (x; λ) are defined by means of the following
gen-erating function:
F A (t, x; λ) = t
λe t– e
tx=
∞
n=
B n (x; λ) t
n
n!.
For x = , we have the Apostol-Bernoulli numbers
B (λ) = B (; λ),
Trang 4B n = B n(; )
which denotes the Bernoulli numbers (of the first kind) (cf [, , –, , , , ],
and the references cited therein)
The λ-Stirling numbers of the second kind are defined by means of the following
gener-ating function:
F LS (t, v; λ) = (λe
t– )v
v! =
∞
n=
S(n, v; λ) t
n
(cf []; see also [, , , , ], and the references cited therein).
In [], the Stirling number of the second kind S(n, k) are defined in combinatorics: the Stirling numbers of the second kind are the number of ways to partition a set of n objects
into k groups These numbers are defined by means of the following generating function:
F S (t, v) = (e
t– )v
v! =
∞
n=
S(n, v) t
n
(cf [–], and the references cited therein) By using the above generating function, these
numbers are computed by the following explicit formula:
S(n, v) =
v!
v
j=
v j
Setting λ = in (.), we have
S(n, v; ) = S(n, v) (cf [–], and the references cited therein).
The Stirling numbers of the first kind s(n, v) are defined by means of the following
gen-erating function:
F s (t, k) = (log( + t))
k
∞
n=
s(n, k) t
n
(cf [, , , , , ], and the references cited therein).
The Bernstein basis functions B n k (x) are defined as follows:
B n k (x) =
n k
x k ( – x) n –k
x ∈ [, ]; n, k ∈ N
where k = , , , n and
n k
k !(n – k)!
(cf [, , , , ]), and the references cited therein.
Trang 5The Bernstein basis functions can also be defined by means of the following generating functions:
f B,k (x, t) = t
k x k e (–x)t
∞
n=
B n k (x) t
n
where k = , , , n and t ∈ C and x ∈ [, ] (cf [, , , ]) and see also the references
cited in each of these earlier works
The Bernoulli polynomials of the second kind b n (x) are defined by means of the
follow-ing generatfollow-ing function:
F b (t, x) = t
log( + t) ( + t)
x=
∞
n=
b n (x) t
n
(cf [], pp.-, and the references cited therein).
The Bernoulli numbers of the second kind b n() are defined by means of the following generating function:
log( + t) =
∞
n=
b n()t
n
n!.
The numbers b n() are known as the Cauchy numbers [, ]
The Daehee polynomials are defined by means of the following generating function:
F D (t, x) =log( + t)
x=
∞
n=
D n (x) t
n
with
D n = D n()
denotes the so-called Daehee numbers (cf [, , , , , , , ], and the references
cited therein)
The Changhee polynomials are defined by means of the following generating function:
F C (t, x) = ( + t)
x
+ t
∞
n=
Ch n (x) t
n
with
Ch n = Ch n()
denotes the so-called Changhee numbers (cf [, , ], and the references cited therein).
Theorem
Zp
x j
dμ(x) =(–)
j
Trang 6Proof of Theorem was given by Schikhof [].
Theorem
Zp
x j
dμ–(x) =(–)
j
j
Proof of Theorem was given by Kim et al [] and [].
Applying the bosonic p-adic integral, the Witt formula for the Daehee numbers and polynomials are given by Kim et al [] as follows, respectively:
D n=
Zp
(x) n dμ(x)
and
D n (y) =
Zp
Applying the fermionic p-adic integral, the Witt formula for the Changhee numbers and polynomials are given by Kim et al [] as follows, respectively:
Ch n=
Zp
(x) n dμ–(x)
and
Ch n (y) =
Zp
Remark Many applications of the fermionic and bosonic p-adic integral onZp have
been given by T Kim and DS Kim first, Jang, Rim, Dolgy, Kwon, Seo, Lim and the others
gave various novel identities, relations and formulas in some special numbers and
poly-nomials (cf [–, , , , , , ], and the references cited therein).
The λ-Bernoulli polynomials B n (x; λ) are defined by means of the following generating
function:
FB(t, x; λ, k) =
log λ + t
λe t–
k
e tx=
∞
n=
B(k) n (x; λ) t
n
(cf []).
In [] and [], Simsek, by using the p-adic Volkenborn integral onZp , defined the
λ-Apostol-Daehee numbers and polynomials, Dn (x; λ) by means of the following generating
functions:
G (t, x; λ) = log λ + log( + λt)
λ ( + λt) – ( + λt)
x=
∞
Dn (x; λ) t
n
Trang 7Observe that substituting λ = into (.), D n (x; ) = D n (x), the Daehee Polynomials (cf.
[, ]) By using (.), we also have the following formula:
D n (x; λ) =
n
k=
n k
(x) n –k D k (λ)
(cf []; see also []).
A relation between the λ-Bernoulli polynomials B n (x; λ), the Apostol-Daehee
polyno-mials Dn (x; λ) and the Stirling numbers of the second kind is given by the following
theo-rem
Theorem
Bm (x; λ) =
m
n=
The proof of (.) was given by the first author in []
Observe that G( e z λ–; λ) is a generating function for the λ-Bernoulli numbers (cf []).
We also observe that G(e z– ; ) is a generating function for the Bernoulli numbers
Substituting λ = into (.), we obtain
G (t; ) =log( + t)
Observe that G(t; ) is a generating function for the Daehee numbers (cf [, ]).
We summarize our results as follows
In Section , we give some identities, relations, and formulas including the Apostol-Daehee numbers and polynomials of higher order, the Changhee numbers and
polynomi-als and the Stirling numbers, the λ-Bernoulli polynomipolynomi-als, the λ-Apostol-Daehee
polyno-mials and the Bernstein basis functions
In Section , we give an integral representation for the Apostol-Daehee polynomials
In Section , we introduce further remarks and observations on these numbers, poly-nomials, and their applications
2 Identities
By using the above generating functions, we get some identities and relation In [] and
[], Simsek gave derivative formulas for the λ-Apostol-Daehee polynomials, D n (x; λ).
Here we give another derivative formula for these polynomials We also give a relation
be-tween the λ-Bernoulli polynomials, the λ-Apostol-Daehee polynomials and the Bernstein
polynomials
Theorem Let n∈ N, then we have
∂
∂xDn+(x; λ) =
n
j=
(–)j j!
n+
j+
λ j+Dn –j (x; λ).
Proof We can take the derivative of equation (.) with respect to x, we obtain the
fol-lowing partial differential equation:
∂
∂x G (t, x; λ) = G(t, x; λ) log( + λt).
Trang 8By using the above partial differential equation with (.), we get
∞
n=
∂
∂xDn (x; λ)
t n
n!=
∞
n=
(–)n λ n+ t n+
n+
∞
n=
Dn (x; λ) t
n
n!. Therefore
∞
n=
∂
∂xDn+(x; λ)
t n
(n + )! =
∞
n=
j=
(–)j λ j+
j+
Dn –j (x; λ) (n – j)!
t n
Comparing the coefficients of t non both sides of the above equation, we arrive at the
Theorem Let n∈ N Starting with
D(x; λ) = log λ
λ– ,
we have
(λ – )
n+ Dn+(x; λ) + λ
Dn (x; λ) = log λ
n+ λ
n+(x) n++ λ n+n!
n
k=
(–)k (x) n –k
(k + )(n – k)!. (.)
Proof By using (.) with the definition of the logarithmic function, we get
(λ – )
∞
n=
n+ Dn+(x; λ)
t n
n! + λ
∞
n=
Dn (x; λ) t
n
n!
= log λ
∞
n=
(x) n
n+
λ n+t n
n! +
∞
n=
(–)n λ
n+t n
n+
∞
n=
(x) n
λ n t n
n! .
By using the Cauchy product in the right side of the above equation, we get
(λ – )
∞
n=
n+ Dn+(x; λ)
t n
n! + λ
∞
n=
Dn (x; λ) t
n
n!
= log λ
∞
n=
(x) n
n+
λ n+t n
n! +
∞
n=
n!
n
k=
(–)k λ n+(x) n –k
(k + )(n – k)!
t n
n!.
After some elementary calculations and comparing the coefficients oft n
n! on both sides of
Setting n = into (.), we compute a few values of the polynomials D n (x; λ) as follows:
D(x; λ) = λ log λ
λ– x+
( – log λ)λ– λ
Theorem
( – y) mBm
x+ y
– y ; λ =
m
B m j (y)
m –j
Trang 9Proof Substituting
t=e
z (–y)–
λ
into (.) and combining with (.), we get the following functional equation:
FB
( – y)z, x + y
– y ; λ, =
∞
m=
y m z
m
m!
∞
n=
λ nDn (x; λ) (e
z (–y)– )n
n! . Combining the above equation with (.), we get
∞
m=
( – y) mB()m
x+ y
– y ; λ
z m
m!
=
∞
m=
y m z
m
m!
∞
n=
λ nDn (x; λ)
∞
m=
S(m, n)( – y) m z
m
m!.
Since n > m, S(m, n) = , we have
∞
m=
( – y) mB()m
x+ y
– y ; λ
z m
m!
=
∞
m=
y m z
m
m!
∞
m=
m
n=
λ nDn (x; λ)S(m, n)( – y) m z
m
m!. Therefore
∞
m=
( – y) mB()m
x+ y
– y ; λ
z m
m!
=
∞
m=
m
j=
B m j (y)
m –j
n=
λ nDn (x; λ)S(m – j, n) z
m
m!.
Comparing the coefficients of z m m! on both sides of the above equation, we arrive at the
Combining (.) with (.), we get the following theorem
Theorem
( – y) mBm
x+ y
– y ; λ =
m
j=
Theorem
( – y) mBm
x+ y
– y ; λ dy=
m+
m
Bm –j (x; λ).
Trang 10Proof Integrating both sides of equation (.) from to with respect to y, we obtain
( – y) mB(k) m
x+ y
– y ; λ dy=
m
j=
Bm –j (x; λ)
B m j (y) dy.
Since
B m j (y) dy =
m+
(cf []), we get
( – y) mB(k) m
x+ y
– y ; λ dy=
m+
m
j=
Bm –j (x; λ).
By using (.), we derive the following functional equation:
G (t, x + y; λ) = ( + λt) y G (t, x; λ).
By using the above functional equations, we have the following theorem
Theorem
Dn (x + y; λ) =
n
j=
n j
3 Integral representation for the Apostol-Daehee polynomials
In [], Simsek defined the Apostol-Daehee polynomials D(k) n (x; λ) of higher order k by
means of the following generating function:
FD(t, x; λ, k) =
log λ + log( + λt)
λt + λ –
k
( + λt) x=
∞
n=
D(k) n (x; λ) t
n
Setting x = in (.) gives the Apostol-Daehee numbers D (k) n (λ) of higher order k:
D(k) n (λ) = D (k) n (; λ).
By the same method as in [], we give a multiple bosonic p-adic integral for the
Apostol-Daehee polynomials of higher order D(k) n (x; λ) in the following form:
Zp
· · ·
Zp
λ x +···+xk ( + λt) x +···+xk +x dμ(x)· · · dμ(x k) =
∞
n=
D(k) n (x; λ) t
n