Research Article Fourier Series of the Periodic Bernoulli and Euler …downloads.hindawi.com/journals/aaa/2014/856491.pdf · 2019. 7. 31. · Research Article Fourier Series of the

Research ArticleFourier Series of the Periodic Bernoulli and Euler Functions

Cheon Seoung Ryoo,1 Hyuck In Kwon,2 Jihee Yoon,2 and Yu Seon Jang3

1 Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea2Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea3 Department of Applied Mathematics, Kangnam University, Yongin 446-702, Republic of Korea

Correspondence should be addressed to Yu Seon Jang; [email protected]

Received 16 April 2014; Revised 16 July 2014; Accepted 16 July 2014; Published 5 August 2014

Academic Editor: Ming Mei

Copyright © 2014 Cheon Seoung Ryoo et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We give some properties of the periodic Bernoulli functions and study the Fourier series of the periodic Euler functions which arederived periodic functions from the Euler polynomials. And we derive the relations between the periodic Bernoulli functions andthose from Euler polynomials by using the Fourier series.

1. Introduction

The numbers and polynomials of Bernoulli and Euler arevery useful in classical analysis and numerical mathemat-ics. Recently, several authors have studied the identitiesof Bernoulli and Euler numbers and polynomials [1–12].The Bernoulli and Euler polynomials, 𝐵

𝑛(𝑥), 𝐸

𝑛(𝑥), 𝑛 =

0, 1, . . ., are defined, respectively, by the following exponentialgenerating functions:

∞

∑

𝑛=0

𝐵𝑛

(𝑥)𝑡𝑛

𝑛!=

𝑡

𝑒𝑡 − 1𝑒𝑥𝑡

,

∞

∑

𝑛=0

𝐸𝑛

(𝑥)𝑡𝑛

𝑛!=

2

𝑒𝑡 + 1𝑒𝑥𝑡

.

(1)

When 𝑥 = 0, these values 𝐵𝑛(0) = 𝐵

𝑛and 𝐸

𝑛(0) = 𝐸

𝑛, 𝑛 =

0, 1, . . ., are called the Bernoulli numbers and Euler numbers,respectively [13].

Euler polynomials are related to the Bernoulli polynomi-als by

𝐸𝑛−1

(𝑥) =2

𝑛{𝐵𝑛

(𝑥) − 2𝑛

𝐵𝑛

(𝑥

2)} ,

𝐸𝑛−2

(𝑥) =4

𝑛 (𝑛 − 1)

𝑛−2

∑

𝑘=0

(𝑛

𝑘) {(2𝑛−𝑘

− 1) 𝐵𝑛−𝑘

𝐵𝑘

(𝑥)} ,

(2)

where (𝑛

𝑘) = 𝑛!/(𝑛 − 𝑘)!𝑘! [3, 14].

Bernoulli polynomials and the related Bernoulli func-tions are of basic importance in theoretical numerical anal-ysis. The periodic Bernoulli functions 𝐵

𝑛(𝑥) are Bernoulli

polynomials evaluated at the fractional part of the argument𝑥 as follows:

𝐵𝑛

(𝑥) = 𝐵𝑛

(⟨𝑥⟩) , (3)

where ⟨𝑥⟩ = 𝑥 − [𝑥] and [𝑥] is the greatest integer lessthan or equal to 𝑥 [13]. Periodic Bernoulli functions play animportant role in several mathematical results such as thegeneral Euler-McLaurin summation formula [1, 10, 15]. Andalso it was shown by Golomb et al. that the periodic Bernoullifunctions serve to construct periodic polynomials splines onuniform meshes. For uniform meshes Delvos showed thatLocher’s method of interpolation by translation is applicableto periodic𝐵-splines.This yields an easy and stable algorithmfor computing periodic polynomial interpolating splines ofarbitrary degree on uniform meshes via Fourier transform[15].

A Fourier series is an expansion of a periodic function𝑓(𝑥) in terms of an infinite sum of sines and cosines. Fourierseries make use of the orthogonality relationships of thesine and cosine functions. Since these functions form a

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 856491, 4 pageshttp://dx.doi.org/10.1155/2014/856491

2 Abstract and Applied Analysis

complete orthogonal system over [−𝜋, 𝜋], the Fourier seriesof a function 𝑓(𝑥) is given by

𝑓 (𝑥) =1

2𝑎0

+

∞

∑

𝑛=1

(𝑎𝑛cos 𝑛𝑥 + 𝑏

𝑛sin 𝑛𝑥) , (4)

where

𝑎0

=1

𝜋∫

𝜋

−𝜋

𝑓 (𝑥) 𝑑𝑥,

𝑎𝑛

=1

𝜋∫

𝜋

−𝜋

𝑓 (𝑥) cos 𝑛𝑥 𝑑𝑥,

𝑏𝑛

=1

𝜋∫

𝜋

−𝜋

𝑓 (𝑥) sin 𝑛𝑥 𝑑𝑥.

(5)

Thenotion of a Fourier series can also be extended to complexcoefficients [16, 17].

The complex form of the Fourier series can be written bythe Euler formula, 𝑒𝑖𝜃 = cos 𝜃 + 𝑖 sin 𝜃 (𝑖 = √−1), as follows:

𝑓 (𝑥) =

∞

∑

𝑘=−∞

𝑐𝑘𝑒𝑖𝑘𝑥

, (6)

where

𝑐𝑘

=1

2𝜋∫

𝜋

−𝜋

𝑓 (𝑥) 𝑒−𝑖𝑘𝑥

𝑑𝑥, 𝑘 ∈ Z. (7)

For a function periodic in [−𝐿/2, 𝐿/2], these become

𝑓 (𝑥) =

∞

∑

𝑛=−∞

𝑐𝑘𝑒𝑖(2𝜋𝑘𝑥/𝐿)

, (8)

where

𝑐𝑘

=1

𝐿∫

𝐿/2

−𝐿/2

𝑓 (𝑥) 𝑒−𝑖(2𝜋𝑘𝑥/𝐿)

𝑑𝑥. (9)

In this paper, we give some properties of the periodicBernoulli functions and study the Fourier series of the peri-odic Euler functions which are derived periodic functionsfrom the Euler polynomials. And we derive the relationsbetween the periodic Bernoulli functions and those fromEuler polynomials by using the Fourier series. We indebtedthis idea to Kim [6–9, 18–20].

2. Periodic Bernoulli and Euler Functions

The periodic Bernoulli functions can be represented asfollows:

𝐵∗

𝑛(𝑥) = 𝐵

𝑛(𝑥) , 0 ≤ 𝑥 < 1, 𝑛 = 0, 1, . . . , (10)

satisfying

𝐵∗

𝑛(𝑥) = 𝐵

∗

𝑛(𝑥 + 1) , 𝑥 ∈ R, 𝑛 = 0, 1, . . . . (11)

From the definition of 𝐵𝑛(𝑥) we know that for 0 ≤ 𝑥 < 1

𝑛

∑

𝑘=0

(𝑛

𝑘) 𝐵∗

𝑘(𝑥) − 𝐵

∗

𝑛(𝑥) = 𝑛𝑥

𝑛−1

, 𝑛 = 0, 1, . . . . (12)

These can be rewritten as follows:

(𝐵∗

(𝑥) + 1)𝑛

− 𝐵∗

𝑛(𝑥) = 𝑛𝑥

𝑛−1

, 𝑛 = 0, 1, . . . , (13)

by using the symbolic convention exhibited by (𝐵∗

(𝑥))𝑛

=:

𝐵∗

𝑛(𝑥) [7].Observe that for 0 ≤ 𝑥 < 1

∞

∑

𝑛=0

𝐵∗

𝑛(1 − 𝑥)

𝑡𝑛

𝑛!=

−𝑡

𝑒−𝑡 − 1𝑒𝑥(−𝑡)

=

∞

∑

𝑛=0

(−1)𝑛

𝐵∗

𝑛(𝑥)

𝑡𝑛

𝑛!. (14)

Since 𝐵∗

𝑛(𝑥), 𝑛 = 0, 1, . . ., are periodic with period 1 onR, we

have

𝐵∗

𝑛(−𝑥) = (−1)

𝑛

𝐵∗

𝑛(𝑥) , 𝑥 ∈ R. (15)

The Apostol-Bernoulli and Apostol-Euler polynomialshave been investigated by many researchers [1, 2, 10, 11]. In[1], Bayad found the Fourier expansion for Apostol-Bernoullipolynomials which are complex version of the classicalBernoulli polynomials. As a result of ordinary Bernoullipolynomials, we have the following lemma.

Lemma 1 (Bayad [1]; see also Luo [10]). The Fourier series of𝐵∗

𝑛(𝑥) on (−1, 1) is

𝐵∗

𝑛(𝑥) = −𝑛! ∑

𝑘∈Z−{0}

1

(2𝜋𝑖𝑘)𝑛𝑒2𝜋𝑖𝑘𝑥

, 𝑛 = 0, 1, . . . . (16)

From Lemma 1 we have the following theorem.

Theorem 2. For |𝑥| < 1 and 𝐿 ∈ N one has

𝐿−1

∑

ℓ=0

𝐵∗

𝑛(

𝑥 + ℓ

𝐿)

=1

𝐿𝐵∗

𝑛(

𝑥

𝐿) − 𝑛! ∑

𝑘∈Z−{0}

[[

[

𝐿−1

∑

ℓ=0

𝑘 ̸≡ 0( mod 𝐿)

𝑒2𝜋𝑖𝑘(𝑥+ℓ)/𝐿

(2𝜋𝑖𝑘)𝑛

]]

]

.

(17)

Proof. Since

𝐿−1

∑

ℓ=0

𝐵∗

𝑛(

𝑥 + ℓ

𝐿) = (−𝑛!)

𝐿−1

∑

ℓ=0

{ ∑

𝑘∈Z−{0}


(2𝜋𝑖𝑘)𝑛

}

= (−𝑛!) ∑

𝑘∈Z−{0}

[[

[

𝐿−1

∑

ℓ=0

𝑘≡0( mod 𝐿)


(2𝜋𝑖𝑘)𝑛

+

𝐿−1

∑

ℓ=0

𝑘 ̸≡ 0( mod 𝐿)


(2𝜋𝑖𝑘)𝑛

]]

]

(18)

and under 𝑘 ≡ 0(mod 𝐿)

𝐿−1

∑

ℓ=0

𝑒2𝜋𝑖ℓ𝑘/𝐿

= 𝐿, (19)

Abstract and Applied Analysis 3

we have

𝐿−1

∑

ℓ=0

𝐵∗

𝑛(

𝑥 + ℓ

𝐿)

= (−𝑛!) ∑

𝑘∈Z−{0}

[[

[

1

𝐿

𝑒2𝜋𝑖𝑘𝑥/𝐿

(2𝜋𝑖𝑘)𝑛

+

𝐿−1

∑

ℓ=0

𝑘 ̸≡ 0( mod 𝐿)


(2𝜋𝑖𝑘)𝑛

]]

]

.

(20)

This implies the desired result.

The periodic Euler polynomials 𝐸𝑛(𝑥) can be usually

defined by 𝐸𝑛(𝑥 + 1) = −𝐸

𝑛(𝑥) and 𝐸

𝑛(𝑥) = 𝐸

𝑛(𝑥) for 0 ≤

𝑥 < 1 [13]. Unlike the periodic Bernoulli, which have period1, the 𝐸

𝑛(𝑥) have periodic 2 and exhibit an even (versus odd)

symmetry about zero [21].As the above Bernoulli case, we consider the periodic

Euler functions as the following:

𝐸∗

𝑛(𝑥) = 𝐸

𝑛(𝑥) , 0 ≤ 𝑥 < 1, 𝑛 = 0, 1, . . . , (21)

such that

𝐸∗

𝑛(𝑥) = 𝐸

∗

𝑛(𝑥 + 1) , 𝑥 ∈ R, 𝑛 = 0, 1, . . . . (22)

Then the functions 𝐸∗

𝑛, 𝑛 = 0, 1, . . ., are also periodic. From

definition of Euler polynomials, we know that

𝑛

∑

𝑘=0

(𝑛

𝑘) 𝐸∗

𝑘(𝑥) + 𝐸

∗

𝑛(𝑥) = 2𝑥

𝑛

, 0 ≤ 𝑥 < 1, 𝑛 = 0, 1, . . . ,

(23)

where 𝐸∗

𝑘= 𝐸𝑘(0) is the 𝑘th Euler number. These can be

rewritten as follows:

(𝐸∗

(𝑥) + 1)𝑛

+ 𝐸∗

𝑛(𝑥) = 2𝑥

𝑛

, 0 ≤ 𝑥 < 1, 𝑛 = 0, 1, . . . ,

(24)

by using the symbolic convention exhibited by (𝐸∗

(𝑥))𝑛

=:

𝐸∗

𝑛(𝑥). When 𝑥 = 0, these relations are given by

(𝐸∗

+ 1)𝑛

+ 𝐸∗

𝑛= 2𝛿0,𝑛

, 𝑛 = 0, 1, . . . , (25)

where 𝛿0,𝑛

is Kronecker symbol and (𝐸∗

+ 1)𝑛 is interpreted

as ∑𝑛

𝑘=0(𝑛

𝑘) 𝐸∗

𝑘[9].

Remark 3. Observe that for 0 ≤ 𝑥 < 1

∞

∑

𝑛=0

𝐸∗

𝑛(1 − 𝑥)

𝑡𝑛

𝑛!=

∞

∑

𝑛=0

(−1)𝑛

𝐸∗

𝑛(𝑥)

𝑡𝑛

𝑛!. (26)

As in the Bernoulli case, we have the following equation:

𝐸∗

𝑛(−𝑥) = (−1)

𝑛

𝐸∗

𝑛(𝑥) , 𝑥 ∈ R. (27)

This means that if 𝑛 is odd (even) number, then 𝐸∗

𝑛is odd

(even) function.

Theorem 4. For |𝑥| < 1/2, one has

𝐸∗

𝑛(𝑥) = −

2

𝑛 + 1𝐸∗

𝑛+1

+ ∑

𝑘∈Z−{0}

[2

𝑛−2

∑

ℓ=1

(𝑛)ℓ𝐸∗

𝑛−ℓ+1

(2𝜋𝑖𝑘)ℓ

(𝑘 − ℓ + 1)

−𝑛!

(2𝜋𝑖𝑘)𝑛] 𝑒2𝜋𝑖𝑘𝑥

.

(28)

Proof. Let

𝐸∗

𝑛(𝑥) =

∞

∑

𝑘=−∞

𝑐∗

𝑘,𝑛𝑒2𝜋𝑖𝑘𝑥 (29)

be the Fourier series for 𝐸∗

𝑛(𝑥), 𝑛 = 0, 1, . . ., on (−1/2, 1/2).

Then

𝑐∗

𝑘,𝑛= ∫

1/2

−1/2

𝐸∗

𝑛(𝑥) 𝑒−2𝜋𝑖𝑘𝑥

𝑑𝑥. (30)

From definition of 𝐸∗

𝑛(𝑥), we have

∫

1/2

−1/2

𝐸𝑛(𝑥)∗

𝑒−2𝜋𝑖𝑘𝑥

𝑑𝑥 = ∫

1

0

𝐸𝑛

(𝑥) 𝑒−2𝜋𝑖𝑘𝑥

𝑑𝑥. (31)

Since 𝐸𝑛+1

(1) = −𝐸𝑛+1

, if 𝑘 = 0, then

𝑐∗

0,𝑛=

1

𝑛 + 1[𝐸𝑛+1

(1) − 𝐸𝑛+1

] = −2

𝑛 + 1𝐸𝑛+1

= −2

𝑛 + 1𝐸∗

𝑛+1.

(32)

For 𝑘 ∈ Z− {0}, so we have the following recurrence relation:

𝑐∗

𝑘,𝑛= −

2

𝑛 + 1𝐸𝑛+1

+2𝜋𝑖𝑘

𝑛 + 1∫

1

0

𝐸𝑛+1

(𝑥) 𝑒−2𝜋𝑖𝑘𝑥

𝑑𝑥

= −2

𝑛 + 1𝐸𝑛+1

+2𝜋𝑖𝑘

𝑛 + 1𝑐∗

𝑘,𝑛+1, 𝑛 = 0, 1, . . . .

(33)

This implies that

𝑐∗

𝑘,𝑛= 2

𝑛−2

∑

ℓ=1

(𝑛)ℓ𝐸𝑛−ℓ+1

(2𝜋𝑖𝑘)ℓ

(𝑛 − ℓ + 1)

+𝑛!

(2𝜋𝑖𝑘)𝑛−1

𝑐∗

𝑘,1,

𝑛 = 0, 1, . . . ,

(34)

where (𝑛)ℓ

= 𝑛(𝑛 − 1) ⋅ ⋅ ⋅ (𝑛 − ℓ + 1) is falling factorial. Since

𝑐∗

𝑘,1= ∫

1

0

(𝑥 −1

2) 𝑒−2𝜋𝑖𝑘𝑥

𝑑𝑥 = −1

2𝜋𝑖𝑘, (35)

we have

𝑐∗

𝑘,𝑛= 2

𝑛−2

∑

ℓ=1

(𝑛)ℓ𝐸𝑛−ℓ+1

(2𝜋𝑖𝑘)ℓ

(𝑛 − ℓ + 1)

+𝑛!

(2𝜋𝑖𝑘)𝑛, 𝑛 = 0, 1, . . . .

(36)

This is completion of the proof.

4 Abstract and Applied Analysis

From Lemma 1 and Theorem 4 we have the followingcorollary.

Corollary 5. For |𝑥| < 1/2 one has

𝐵∗

𝑛(𝑥) = 𝐸

∗

𝑛(𝑥) +

2

𝑛 + 1𝐸∗

𝑛+1

− 2 ∑

𝑘∈Z−{0}

[

𝑛−2

∑

ℓ=1

(𝑛)ℓ𝐸∗

𝑛−ℓ+1

(2𝜋𝑖𝑘)𝑙

(𝑛 − ℓ + 1)

] 𝑒2𝜋𝑖𝑘𝑥

,

(37)

where 𝐸∗

𝑛−ℓ+1= 𝐸∗

𝑛−ℓ+1(0) and (𝑛)

ℓ= 𝑛(𝑛 − 1) ⋅ ⋅ ⋅ (𝑛 − ℓ + 1) is

falling factorial.

Corollary 6. For |𝑥| < 1/2 one has

𝐵∗

𝑛(𝑥) = 𝐸

∗

𝑛(𝑥) +

2

𝑛 + 1𝐸∗

𝑛+1+ 2

𝑛−2

∑

ℓ=1

(𝑛

ℓ)

𝐸∗

𝑛−ℓ+1

𝑛 − ℓ + 1𝐵∗

ℓ(𝑥) ,

(38)

where 𝐸∗

𝑛−ℓ+1= 𝐸∗

𝑛−ℓ+1(0).

Proof. From Lemma 1, we have

∑

𝑘∈Z−{0}

[

𝑛−2

∑

ℓ=1

(𝑛)ℓ𝐸∗

𝑛−ℓ+1

(2𝜋𝑖𝑘)ℓ

(𝑛 − ℓ + 1)

] 𝑒2𝜋𝑖𝑘𝑥

=

𝑛−2

∑

ℓ=1

(𝑛

ℓ)

𝐸∗

𝑛−ℓ+1

(𝑛 − ℓ + 1)[ℓ! ∑

𝑘∈Z−{0}

𝑒2𝜋𝑖𝑘𝑥

(2𝜋𝑖𝑘)ℓ]

=

𝑛−2

∑

ℓ=1

(𝑛

ℓ)

𝐸∗

𝑛−ℓ+1

(𝑛 − ℓ + 1)(−𝐵∗

ℓ(𝑥)) .

(39)

This becomes the desired result.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors would like to thank T. Kim for all the motivationand insightful conversations on this subject. The authorswould also like to thank the referee(s) of this paper for thevaluable comments and suggestions.

References

[1] A. Bayad, “Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials,” Mathematics ofComputation, vol. 80, no. 276, pp. 2219–2221, 2011.

[2] D. Ding and J. Yang, “Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials,” Advanced Studies inContemporary Mathematics, vol. 20, no. 1, pp. 7–21, 2010.

[3] K. W. Hwang, D. V. Dolgy, D. S. Kim, T. Kim, and S. H.Lee, “Some theorems on Bernoulli and Euler numbers,” ArsCombinatoria, vol. 109, pp. 285–297, 2013.

[4] J. H. Jeong, J. H. Jin, J. W. Park, and S. H. Rim, “On thetwisted weak q-Euler numbers and polynomials with weight 0,”Proceedings of the Jangjeon Mathematical Society, vol. 16, no. 2,pp. 157–163, 2013.

[5] D. S. Kim, N. Lee, J. Na, and K. H. Park, “Identities of symmetryfor higher-order Euler polynomials in three variables (I),”Advanced Studies in Contemporary Mathematics, vol. 22, no. 1,pp. 51–74, 2012.

[6] T. Kim, “Identities involving Frobenius-Euler polynomials aris-ing from non-linear differential equations,” Journal of NumberTheory, vol. 132, no. 12, pp. 2854–2865, 2012.

[7] T. Kim, J. Choi, andY.H.Kim, “Anote on the values of Euler zetafunctions at posive integers,”Advanced Studies in ContemporaryMathematics, vol. 22, no. 1, pp. 27–34, 2012.

[8] T. Kim, D. S. Kim, D. V. Dolgy, and S. H. Rim, “Some identitieson the Euler numbers arising fromEuler basis polynomials,”ArsCombinatoria, vol. 109, pp. 433–446, 2013.

[9] T. Kim, B. Lee, S. H. Lee, and S. H. Rim, “Identities for theBernoulli and Euler numbers and polynomials,” Ars Combina-toria, vol. 107, pp. 325–337, 2012.

[10] Q. Luo, “Fourier expansions and integral representations for theApostol-Bernoulli and Apostol-Euler polynomials,”Mathemat-ics of Computation, vol. 78, no. 268, pp. 2193–2208, 2009.

[11] E. Sen, “Theorems on Apostol-Euler polynomials of higherorder arising from Euler basis,” Advanced Studies in Contempo-rary Mathematics, vol. 23, no. 2, pp. 337–345, 2013.

[12] Y. Simsek, O. Yurekli, and V. Kurt, “On interpolation functionsof the twisted generalized Frobenius-Euler numbers,”AdvancedStudies in ContemporaryMathematics, vol. 15, no. 2, pp. 187–194,2007.

[13] M. Abramowitz and J. Stegun, Handbook of MathematicalFunctions with Formulas, Graphs, and Mathematical Tables,Dover, New York, NY, USA, 1972.

[14] M. Aigner, Combinatorial Theory, vol. 234 of GrundlehrenderMathematischenWissenschaften, Springer, Berlin, Germany,1979.

[15] F. Delvos, “Bernoulli functions and periodic 𝐵-splines,” Com-puting, vol. 38, no. 1, pp. 23–31, 1987.

[16] R. Askey and D. T. Haimo, “Similarities between Fourier andpower series,”TheAmericanMathematicalMonthly, vol. 103, no.4, pp. 297–304, 1996.

[17] G. B. Folland, Fourier analysis and Its applications, Wadsworth& Brooks Cole Advanced Books & Software, Pacific Grove,Calif, USA, 1992.

[18] T. Kim, “Euler numbers and polynomials associated with zetafunctions,” Abstract and Applied Analysis, vol. 2008, Article ID581582, 11 pages, 2008.

[19] T. Kim, “On Euler-Barnes multiple zeta functions,” RussianJournal of Mathematical Physics, vol. 10, no. 3, pp. 261–267, 2003.

[20] T. Kim, “Note on the Euler numbers and polynomials,”Advanced Studies in Contemporary Mathematics, vol. 17, no. 2,pp. 131–136, 2008.

[21] J. M. Borwein, N. J. Calkin, and D. Manna, “Euler-Boolesummation revisited,” The American Mathematical Monthly,vol. 116, no. 5, pp. 387–412, 2009.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

Documents

Research Article Fourier Series of the Periodic Bernoulli and Euler …downloads.hindawi.com/journals/aaa/2014/856491.pdf · 2019. 7. 31. · Research Article Fourier Series of the