Research Article | Open Access
Fourier Series of the Periodic Bernoulli and Euler Functions
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.
The numbers and polynomials of Bernoulli and Euler are very useful in classical analysis and numerical mathematics. Recently, several authors have studied the identities of Bernoulli and Euler numbers and polynomials [1–12]. The Bernoulli and Euler polynomials, , , , are defined, respectively, by the following exponential generating functions: When , these values and , , are called the Bernoulli numbers and Euler numbers, respectively .
Bernoulli polynomials and the related Bernoulli functions are of basic importance in theoretical numerical analysis. The periodic Bernoulli functions are Bernoulli polynomials evaluated at the fractional part of the argument as follows: where and is the greatest integer less than or equal to . 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 .
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, , as follows: where For a function periodic in , 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 [6–9, 18–20].
2. Periodic Bernoulli and Euler Functions
The periodic Bernoulli functions can be represented as follows: satisfying From the definition of we know that for These can be rewritten as follows: by using the symbolic convention exhibited by .
Observe that for Since , , are periodic with period on , we have
The Apostol-Bernoulli and Apostol-Euler polynomials have been investigated by many researchers [1, 2, 10, 11]. In , 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.
From Lemma 1 we have the following theorem.
Theorem 2. For and one has
Proof. Since and under 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 , , are also periodic. From definition of Euler polynomials, we know that where is the th Euler number. These can be rewritten as follows: by using the symbolic convention exhibited by . When , these relations are given by where is Kronecker symbol and is interpreted as .
Remark 3. Observe that for As in the Bernoulli case, we have the following equation: This means that if is odd (even) number, then is odd (even) function.
Theorem 4. For , one has
Proof. Let be the Fourier series for , , on . Then From definition of , we have Since , if , then For , so we have the following recurrence relation: This implies that where is falling factorial. Since we have This is completion of the proof.
Corollary 5. For one has where and is falling factorial.
Corollary 6. For one has where .
Proof. From Lemma 1, we have This becomes the desired result.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank T. Kim for all the motivation and insightful conversations on this subject. The authors would also like to thank the referee(s) of this paper for the valuable comments and suggestions.
- A. Bayad, “Fourier expansions for Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials,” Mathematics of Computation, vol. 80, no. 276, pp. 2219–2221, 2011.
- D. Ding and J. Yang, “Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 1, pp. 7–21, 2010.
- K. W. Hwang, D. V. Dolgy, D. S. Kim, T. Kim, and S. H. Lee, “Some theorems on Bernoulli and Euler numbers,” Ars Combinatoria, vol. 109, pp. 285–297, 2013.
- J. H. Jeong, J. H. Jin, J. W. Park, and S. H. Rim, “On the twisted weak q-Euler numbers and polynomials with weight 0,” Proceedings of the Jangjeon Mathematical Society, vol. 16, no. 2, pp. 157–163, 2013.
- D. S. Kim, N. Lee, J. Na, and K. H. Park, “Identities of symmetry for higher-order Euler polynomials in three variables (I),” Advanced Studies in Contemporary Mathematics, vol. 22, no. 1, pp. 51–74, 2012.
- T. Kim, “Identities involving Frobenius-Euler polynomials arising from non-linear differential equations,” Journal of Number Theory, vol. 132, no. 12, pp. 2854–2865, 2012.
- T. Kim, J. Choi, and Y. H. Kim, “A note on the values of Euler zeta functions at posive integers,” Advanced Studies in Contemporary Mathematics, vol. 22, no. 1, pp. 27–34, 2012.
- T. Kim, D. S. Kim, D. V. Dolgy, and S. H. Rim, “Some identities on the Euler numbers arising from Euler basis polynomials,” Ars Combinatoria, vol. 109, pp. 433–446, 2013.
- T. Kim, B. Lee, S. H. Lee, and S. H. Rim, “Identities for the Bernoulli and Euler numbers and polynomials,” Ars Combinatoria, vol. 107, pp. 325–337, 2012.
- Q. Luo, “Fourier expansions and integral representations for the Apostol-Bernoulli and Apostol-Euler polynomials,” Mathematics of Computation, vol. 78, no. 268, pp. 2193–2208, 2009.
- E. Sen, “Theorems on Apostol-Euler polynomials of higher order arising from Euler basis,” Advanced Studies in Contemporary Mathematics, vol. 23, no. 2, pp. 337–345, 2013.
- Y. Simsek, O. Yurekli, and V. Kurt, “On interpolation functions of the twisted generalized Frobenius-Euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 187–194, 2007.
- M. Abramowitz and J. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, NY, USA, 1972.
- M. Aigner, Combinatorial Theory, vol. 234 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1979.
- F. Delvos, “Bernoulli functions and periodic -splines,” Computing, vol. 38, no. 1, pp. 23–31, 1987.
- R. Askey and D. T. Haimo, “Similarities between Fourier and power series,” The American Mathematical Monthly, vol. 103, no. 4, pp. 297–304, 1996.
- G. B. Folland, Fourier analysis and Its applications, Wadsworth & Brooks Cole Advanced Books & Software, Pacific Grove, Calif, USA, 1992.
- T. Kim, “Euler numbers and polynomials associated with zeta functions,” Abstract and Applied Analysis, vol. 2008, Article ID 581582, 11 pages, 2008.
- T. Kim, “On Euler-Barnes multiple zeta functions,” Russian Journal of Mathematical Physics, vol. 10, no. 3, pp. 261–267, 2003.
- T. Kim, “Note on the Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 131–136, 2008.
- J. M. Borwein, N. J. Calkin, and D. Manna, “Euler-Boole summation revisited,” The American Mathematical Monthly, vol. 116, no. 5, pp. 387–412, 2009.
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