Fourier Series of the Periodic Bernoulli and Euler Functions

and Applied Analysis 3 we have

Euler polynomials are related to the Bernoulli polynomials by where Bernoulli polynomials and the related Bernoulli functions are of basic importance in theoretical numerical analysis.The periodic Bernoulli functions B () are Bernoulli polynomials evaluated at the fractional part of the argument  as follows: where ⟨⟩ =  − [] and [] is the greatest integer less than or equal to  [13].Periodic Bernoulli functions play an important role in several mathematical results such as the general Euler-McLaurin summation formula [1,10,15].And also it was shown by Golomb et al. that the periodic Bernoulli functions serve to construct periodic polynomials splines on uniform meshes.For uniform meshes Delvos showed that Locher's method of interpolation by translation is applicable to periodic -splines.This yields an easy and stable algorithm for computing periodic polynomial interpolating splines of arbitrary degree on uniform meshes via Fourier transform [15].A Fourier series is an expansion of a periodic function () in terms of an infinite sum of sines and cosines.Fourier series make use of the orthogonality relationships of the sine and cosine functions.Since these functions form a complete orthogonal system over [−, ], the Fourier series of a function () is given by where The notion of a Fourier series can also be extended to complex coefficients [16,17].
The complex form of the Fourier series can be written by the Euler formula,   = cos  +  sin  ( = √ −1), as follows: where For a function periodic in [−/2, /2], these become where In this paper, we give some properties of the periodic Bernoulli functions and study the Fourier series of the periodic Euler functions which are derived periodic functions from the Euler polynomials.And we derive the relations between the periodic Bernoulli functions and those from Euler polynomials by using the Fourier series.We indebted this idea to Kim [6][7][8][9][18][19][20].

Periodic Bernoulli and Euler Functions
The periodic Bernoulli functions can be represented as follows: satisfying From the definition of   () we know that for 0 ≤  < 1 These can be rewritten as follows: by using the symbolic convention exhibited by ( * ())  =:  *  () [7].
Observe that for 0 ≤  < 1 Since  *  (),  = 0, 1, . .., are periodic with period 1 on R, we have The Apostol-Bernoulli and Apostol-Euler polynomials have been investigated by many researchers [1,2,10,11].In [1], Bayad found the Fourier expansion for Apostol-Bernoulli polynomials which are complex version of the classical Bernoulli polynomials.As a result of ordinary Bernoulli polynomials, we have the following lemma.
Theorem 2. For || < 1 and  ∈ N one has Proof.Since and under  ≡ 0(mod ) we have This implies the desired result.
As the above Bernoulli case, we consider the periodic Euler functions as the following: such that Then the functions  *  ,  = 0, 1, . .., are also periodic.From definition of Euler polynomials, we know that where  *  =   (0) is the th Euler number.These can be rewritten as follows: by using the symbolic convention exhibited by ( * ())  =:  *  ().When  = 0, these relations are given by where  0, is Kronecker symbol and ( * + 1)  is interpreted as ∑  =0 (   )  *  [9].
From Lemma 1 and Theorem 4 we have the following corollary.
This becomes the desired result.