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R E S E A R C H

Open Access

Construction of Fourier expansion of

Apostol Frobenius–Euler polynomials and its

applications

Serkan Araci

1*

and Mehmet Acikgoz

2

*Correspondence:

mtsrkn@hotmail.com

1Department of Economics, Faculty

of Economics, Administrative and Social Science, Hasan Kalyoncu University, Gaziantep, Turkey Full list of author information is available at the end of the article

Abstract

In the present paper, we find the Fourier expansion of the Apostol Frobenius–Euler polynomials. By using a Fourier expansion of the Apostol Frobenius–Euler

polynomials, we derive some new and interesting results.

MSC: Primary 11B68; 11S80; secondary 05A19; 42B05

Keywords: Fourier series; Apostol Frobenius–Euler polynomials; Generating function; Lipschitz summation formula; Hurwitz zeta type function

1 Introduction

When Fourier was trying to solve a problem in heat conduction, he needed to express a functionf as an infinite series ofsineandcosinefunctions:

f(x) =a0 2 +

n=1

(ancosnx+bnsinnx).

Earlier, Bernoulli and Euler had used such series while investigating problems concerning vibrating strings and astronomy. Note that a Fourier series is widely known as an expansion of a periodic functionf(x) in terms of an infinite sum ofsineandcosinefunctions. Fourier series make use of the orthogonality relationships of thesineandcosinefunctions.

Fourier series of a function with period T can be written in an exponential form as follows:

f(x) =a0+a1eiwx+a2e2iwx+· · ·+aneniwx+· · ·

+a–1eiwx+a–2e–2iwx+· · ·+aneniwx+· · ·

or equivalently by

f(x) =

n=–∞

aneinwx

w=2π

T

, (1)

(2)

where the coefficientsanandanare computed by

an=

1

T 2π

w

0

einwtf(t)dt and an=

1

T 2π

w

0

einwtf(t)dt.

Note thatan=an(anis the complex conjugate ofan). For more information as regards

Fourier series, see Refs. [3, 4, 6].

The Fourier expansion of some well-known polynomials have been studied by some mathematicians; see, for details, [2–4, 6]. For example, in [3], Luo derived a Fourier se-ries and integral representations for the classical Genocchi polynomials, and Apostol– Genocchi polynomials by using the Lipschitz summation formula. In [4], by making use of Cauchy residue theorem in the complex plane, Bayad obtained a Fourier series for the Apostol–Bernoulli, Apostol–Genocchi and Apostol–Euler polynomials. Also the Fourier series of sums of products of some well-known special polynomials have been investigated extensively by Agarwalet al.[1] and Kimet al.[6–8, 11].

With this motivation, we are going to focus on obtaining the Fourier expansion of the Apostol Frobenius–Euler polynomial. After that, we are going to derive some useful results arising from its Fourier expansion. Before giving the main results mentioned above, we need some useful properties of Apostol–Frobenius Euler polynomials, which will be given in the next section as preliminaries.

2 Preliminaries

The Frobenius–Euler polynomials and their various generalizations such as the Apostol Frobenius–Euler polynomials have been studied intensively by some mathematicians. For example, Kim [2] obtained linear differential equations for Frobenius–Euler polynomials by using their generating function. From those differential equations, he gave the sums of products of Frobenius–Euler polynomials.

We now begin some known definitions and properties of Frobenius–Euler polynomials and Apostol Frobenius–Euler polynomials which will be useful in deriving the main results of this paper.

Definition 1 Letu∈Cwithu= 1. The Frobenius–Euler polynomialsHn(x,u) are known

as [2, 5]:

n=0

Hn(x,u) tn n! =

1 –u etue

xt.

Definition 2 Letu,λ∈Cwithu= 1,λ= 1 andu=λ. The Apostol Frobenius–Euler poly-nomialsHn(x,u,λ) are defined by [2, 5]

n=0

Hn(x,u,λ) tn n!=

1 –u λetue

xt.

Note that Apostol Frobenius–Euler polynomials are a good generalization (or known as

λextension) of Frobenius–Euler polynomials. Comparing Definition 1 with Definition 2, one may get the following:

Hn(x,u,λ) = u λ

1 –u λ Hn

(3)

Observe that

Hn(0,u) :=Hn(u) and Hn(0,u,λ) :=Hn(u,λ),

which are called the Frobenius–Euler numbers and Apostol Frobenius–Euler numbers, respectively.

Definition 3 The Frobenius–Genocchi polynomials are defined by means of the follow-ing generatfollow-ing function:

n=0

GFn(x,u)t

n

n!= (1 –u)t

etu e

xt cf. [12].

The Apostol–Euler polynomials are defined by means of the following generating series:

n=0

En(x,λ) tn n!=

2

λet+ 1e

xt |t|<log(–λ). (2)

From Eq. (2), we have

2

λet+ 1e

xt= 2λ–1 et+λ–1e

xt= 2

1 +λ

1 +λ–1 et+λ–1e

xt=

n=0 2 1 +λHn

x,λ–1t

n

n!,

from which one may get the following useful corollary.

Corollary 1

En(x,λ) =

2 1 +λHn

x,λ–1, cf.[9]. (3)

Remark1 Substitutingλ= 1 in Definition 2, one can easily see that

H(x,u, 1) :=H(x,u), cf. [2, 5].

Remark2 Puttingu= –1 in Definition 2,

H(x, –1,λ) :=En(x,λ),

whereEn(x,λ) is called the Apostol–Euler polynomials, cf. [2, 5].

Remark3 Puttingu= –1 in Definition 2,

H(x, –1,λ) :=Gn+1(x,λ)

n+ 1 ,

(4)

Remark4 Takingu= –1 andλ= 1 in Definition 2, one can see

H(x, –1, 1) :=En(x),

whereEn(x) is called the classical Euler polynomials, cf. [2, 5].

Remark5 Takingu= –1 andλ= 1 in Definition 2, one can see

H(x, –1, 1) :=Gn+1(x)

n+ 1 ,

whereGn(x) is called the classical Genocchi polynomials, cf. [2, 5].

Proposition 1 The following identity holds true:

λHn(x+ 1,u,λ) –uHn(x,u,λ) = (1 –u)xn, cf.[5].

Proof It is proved by using Definition 2 as follows:

n=0

λHn(x+ 1,u,λ) –uHn(x,u,λ)

tn n!=λ

1 –u λetue

(x+1)tu 1 –u λetue

xt

= (1 –u)ext

=

n=0

(1 –u)xnt n

n!.

Matching the coefficients tnn! gives the required result.

Proposition 2 The following identity holds true:

d

dxHn(x,u,λ) =nHn–1(x,u,λ), cf.[5].

Proof From Definition 2, we have

d dx

n=0

Hn(x,u,λ) tn n!

=

n=0

d

dxHn(x,u,λ)

tn n!

= d

dx

1 –u λetue

xt

= 1 –u

λetute xt

=

n=0

Hn(x,u,λ) tn+1

n! .

Comparing the coefficientstnyields the desired result.

Proposition 3 Let n be a member of the natural numbers.Then we have

1

0

Hn(x,u,λ)dx= uλ

λ

Hn+1(u,λ)

(5)

Proof From Definition 2 and Proposition 1, we have

1

0

Hn(x,u,λ)dx=

Hn+1(1,u,λ) –Hn+1(u,λ)

n+ 1

=uλ

λ

Hn+1(u,λ)

n+ 1 .

Thus we complete the proof of the theorem.

We are now in a position to state our main results in the next section. Also we derive their special cases.

3 Main results

We begin with the following theorem, which is a Fourier series expansion of the Apostol Frobenius–Euler polynomial. For the following theorem, we will give two proofs. The first proof includes the Cauchy residue theorem and a complex integral over a circleC follow-ing Bayad’s method in [4]. The second proof includes the Lipschitz summation formula following Luo’s method in [3].

Theorem 1 Let u,λ∈Cwith u= 1,λ= 1,u=λand0 <x< 1.We have

Hn(x,u,λ) = u– 1

u n!

u λ

x

k∈Z

e2kπix

(2πik–log(λ u))n+1

.

Proof1of Theorem1 We firstly consider the following integral and the functionfn(t) =

1

tn+1λ1–etuuext:

C

fn(t)dt (4)

over the circleC={t| |t| ≤(2N+ε)πandε∈R, (επi±log(λ

u)= 0(mod2πi))}. Now we

find the poles of the functionfn(t) as follows:

tk= 2kπi–log

λ u

(k∈Z)

andt= 0 is a pole of ordern+ 1. From the Cauchy residue theorem, we write

C

fn(t)dt= 2πi

Resfn(t),t= 0

+

k∈Z

Resfn(t),t=tk

. (5)

We should computeRes(fn(t),t= 0) andRes(fn(t),t=tk) as follows:

Resfn(t),t= 0

=lim t→0

1

n!

dn dtn(t– 0)

n+1 1

tn+1

m=0

Hm(x,u,λ) tm m!

=lim t→0

1

n!

dn dtn

m=0

(6)

=lim

Combining these residues with Eq. (5) yields

Therefore, we complete the proof.

Before giving the second proof of Theorem 1, we need the following definition.

Definition 4 Lipschitz summation formula is defined by

plane;denotes the Euler–Gamma function; cf. [3].

(7)

=u– 1

Differentiatingntimes with respect toτgives

From this, we reach the following expression:

which is the desired result.

Corollary 2 From Remark4and Theorem1,we have

(8)

Proof It follows from Remark 4 and Theorem 1 that

Hn(x, –1, 1) :=En(x) = 2(–1)xn!

k∈Z

e2πikx

(2πik–log(–1))n+1.

Since

(–1)x:=eπix and log(–1) = –πi

with the logarithmic property over the complex plane,

z=|z|eiθ, –πθ<π and log(z) =log|z|+,

we can write

En(x) = 2n!

k∈Z

e2πikxπix

(2πik+πi)n+1

= 2n! (2πi)n+1

k∈Z

e2πi(k–12) (k+12)n+1,

which completes the proof of this corollary.

Corollary 3 By making use of the relation H(x, –1, 1) :=Gn+1(x)

n+1 in Theorem1,we have

Gn(x) =

2n! (2πi)n

k∈Z

e2πi(k+12)x

(k+12)n, cf.[4].

Corollary 4 Puttingλ= 1in Theorem1we have

Hn(x,u) = u– 1

u n!u

x

k∈Z

e2kxπi

(2kπi–log(u))n+1.

Corollary 5 Substituting u= –1in Theorem1yields

En(x,λ) =

2n!

λx

k∈Z

e2πi(k–12)x

(2kπiπi–log(λ))n+1, cf.[3, 4].

From a Fourier expanison of the Apostol Frobenius–Euler polynomials, we derive the following interesting identity.

Theorem 2 Let L be a positive integer.Then we have

1

L L–1

j=0

λ u

j L

Hn

x+j

L ,u,λ

=Hn

x L,u,λ

+1

L u– 1

u

u λ

x n!

k∈Z

⎛ ⎜ ⎜ ⎝

L–1

j=0

k=0(modL)

e2πik(xL+j) (2πik–log(λu))n+1

(9)

Proof From Theorem 1, we derive the following applications:

Under the following condition:

L–1

Thus we complete the proof of the theorem.

Theorem 3 Let0 <x< 1.We have

Hn(x,u,λ) =

k∈Z

ck,ne2kπix

with the following the coefficients ck,n:

(10)

with the following coefficients:

ck,n=

1

0

Hn(x,u,λ)e–2πikxdx.

By integration by parts, we have

ck,n= uλ

λ

Hn+1(u,λ)

n+ 1 + 2πik n+ 1

1

0

Hn+1(x,u,λ)e–2πikxdx

=uλ

λ

Hn+1(u,λ)

n+ 1 + 2πik n+ 1ck,n+1.

From this, we find the following recurrence relation:

ck,n= n

2πik λu

λ

Hn(u,λ)

n +

n

2πikck,n–1.

By the iteration method, we arrive at the following expression:

ck,n= λu

λ n–2

l=1

(n)lHnl+1(u,λ) (2πik)l(nl+ 1)+

n! (2πik)n–1ck,1.

Now it seems to be sufficient in order to computeck,1. Since

H1(x,u,λ) = 1 –u λux

λ(1 –u) (λu)2,

we have

ck,1=

1

0

1 –u λux

λ(1 –u) (λu)2

e–2πikxdx

=u– 1

λu

1 2πik.

Also

ck,n= λu

λ n–2

l=1

(n)lHnl+1(u,λ) (2πik)l(nl+ 1)+

1 –u λu

n! (2πik)n.

Thus we end the proof.

In [9], Kimet al.defined the Hurwitz typeλ-zeta function as follows:

ζλ(s,x) =

n=0

λn

(n+x)s (s∈C). (9)

Note thatζλ(s,x) whenλ= –1 is the Hurwitz–Euler zeta function; cf. [10].

Recall from Eq. (8) that

k=0

Hk(x,u,λ) tn n! =

u– 1

u

k=0

λ u

k

e(k+x)t λ ue

t< 1

(11)

From this we have

Hn(x,u,λ) = u– 1

u

k=0

λ u

k

(k+x)n.

Then Eq. (9) can be written

u– 1

u ζλu(–n,x) =

u– 1

u

k=0 (k+x)n

λ u

k

=Hn(x,u,λ).

Thus we have the following theorem.

Theorem 4 The following equality holds true:

u– 1

u ζλu(–n,x) =Hn(x,u,λ).

In [9], Kimet al.introduced theλ-partial zeta function as follows:

Hλ(s,a|F) =

ma(modF)

λm ms.

From this we have the following applications:

Hλ

u(s,a|F) =

m=0

(λu)mF+a

(mF+a)s

=λaFs

m=0 ((λ

u) F)m

(m+Fa)s

=λaFsζ

λF

uF

s,a

F

. (10)

By Theorem 4 and Eq. (10), we have the following theorem.

Theorem 5 The following identity holds true:

λaFnHn

a F,u

F,λF

=u

F– 1

uF Hλu(–n,a|F).

Setλ=e2πix,x= –log(λu)

2πi ands=n+ 1 in Eq. (9), we see that

ζe2πix

n+ 1, –log(

λ u)

2πi

=

k=0

e2πikx

(k–log(λu)

2πi )n+1

(12)

Now we write the Fourier expansion of the Apostol Frobenius–Euler polynomials as

fol-which is closely related to Eq. (11). So we have

Hn(x,u,λ) =

Thus we state the following theorem.

Theorem 6 Let u,λ∈Cwith u= 1,λ= 1,u=λand0 <x< 1.We have

4 Further remarks

Based on Definition 3, we introduce here the Apostol Frobenius–Genocchi polynomials

GF

n(x,u,λ) by the following definition.

Definition 5 Letu∈Cwithu= 1. We define the Apostol Frobenius–Genocchi

polyno-mials as follows:

The Apostol Frobenius–Genocchi polynomials are closely related to the Apostol Frobenius–Euler polynomials by the following application:

(13)

We now give some of fundamental properties of the Apostol Frobenius–Genocchi poly-nomials. We will omit the proof, since it follows from Definition 5.

Theorem 7 The derivative property of the Apostol Frobenius–Genocchi polynomials is as follows:

d dxG

F

n(x,u,λ) =nGFn–1(x,u,λ).

Theorem 8 Difference property of Apostol Frobenius–Genocchi polynomials is as follows:

λGFn(x+ 1,u,λ) –uGFn(x,u,λ) = (1 –u)nxn–1.

Theorem 9 The integral of Apostol Frobenius–Genocchi polynomials from a to b,where a,b are members of real numbers,is as follows:

b

a

GFn(x,u,λ)dx=G

F

n+1(b,u,λ) –GFn+1(a,u,λ)

n+ 1 .

Theorem 10 For|λet

u |< 1,the generating function of the Apostol Frobenius–Genocchi poly-nomials can be written in the following form:

n=0

GFn(x,u,λ)t

n

n! =

u– 1

u te xt

m=0

λ u

m et(m+x).

Theorem 11 By the relation Hn(x,u,λ) = GF

n+1(x,u,λ)

n+1 in Theorem1,we have

GFn(x,u,λ) =u– 1

u

u λ

x n!

k∈Z

e2πikx

(2πik–log(λu))n

which represents a Fourier expansion of the Apostol Frobenius–Genocchi polynomials.

5 Conclusion and observation

In the paper, we have derived the Fourier expansion of Apostol Frobenius–Euler polyno-mials as Theorem 1. We have investigated special cases of Theorem 1 turning to Fourier expansions of Euler polynomials, Genocchi polynomials, Frobenius–Euler polynomials, Apostol–Euler polynomials, Apostol Genocchi polynomials. With the motivation of the work [12], we have introduced Apostol Frobenius–Genocchi polynomials. We saw that Apostol Frobenius–Genocchi polynomials are closely related to Apostol Frobenius–Euler polynomials by the following relation:

Hn(x,u,λ) = GF

n+1(x,u,λ)

n+ 1 . (12)

By this relation, we have got a Fourier expansion for the Apostol Frobenius–Genocchi polynomials,

GFn(x,u,λ) =u– 1

u

u λ

x n!

k∈Z

e2πikx

(14)

By Eq. (12), the obtained theorems concerning the Apostol Frobenius–Euler polynomi-als here and in other sources turn into those concerning Apostol Frobenius–Genocchi polynomials.

Acknowledgements

The authors are grateful to the Editor, Prof. Dr. Taekyun Kim, and to the referees for their valuable suggestions, which have improved the paper substantially.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript. All authors read and approved the final manuscript.

Author details

1Department of Economics, Faculty of Economics, Administrative and Social Science, Hasan Kalyoncu University,

Gaziantep, Turkey. 2Department of Mathematics, Faculty of Science and Arts, University of Gaziantep, Gaziantep, Turkey.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 22 December 2017 Accepted: 15 February 2018

References

1. Agarwal, R.P., Kim, D.S., Kim, T., Kwon, J.: Sums of finite products of Bernoulli functions. Adv. Differ. Equ.2017, 237 (2017)

2. Kim, T.: Identities involving Frobenius–Euler polynomials arising from non-linear differential equations. J. Number Theory132, 2854–2865 (2012)

3. Luo, Q.-M.: Fourier expansions and integral representations for the Apostol–Bernoulli and Apostol–Euler polynomials. Math. Comput.78(268), 2193–2208 (2009)

4. Bayad, A.: Fourier expansions for Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials. Math. Comput.80(276), 2219–2221 (2011)

5. Bayad, A., Kim, T.: Identities for Apostol-type Frobenius–Euler polynomials resulting from the study of a nonlinear operator. Russ. J. Math. Phys.23(2), 164–171 (2016)

6. Kim, T., Kim, D.S., Jang, G.-W., Kwon, J.: Fourier series of sums of products of Genocchi functions and their applications. J. Nonlinear Sci. Appl.10, 1683–1694 (2017)

7. Jang, G.-W., Kim, T., Kim, D.S., Mansour, T.: Fourier series of functions related to Bernoulli polynomials. Adv. Stud. Contemp. Math.27, 49–62 (2017)

8. Kim, T., Kim, D.S., Rim, S.-H., Dolgy, D.: Fourier series of higher-order Bernoulli functions and their applications. J. Inequal. Appl.2017, 8 (2017)

9. Kim, T., Rim, S.-H., Simsek, Y., Kim, D.: On the analogs of Bernoulli and Euler numbers, related identities and zeta and

L-functions. J. Korean Math. Soc.45(2), 435–453 (2008)

10. Kim, T.: Euler numbers and polynomials associated with zeta functions. Abstr. Appl. Anal.2008, Article ID 581582 (2008)

11. Kim, T., Kim, D.S., Dolgy, D.V., Park, J.-W.: Fourier series of sums of products of ordered Bell and poly-Bernoulli functions. J. Inequal. Appl.2017, 84 (2017)

References

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