Taekyun Kim, Dae San Kim, Toufik Mansour and Gwan-Woo Jang

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 24, 2018, Number 1, Pages 76—93

DOI: 10.7546/nntdm.2018.24.1.76-93

**Download full paper: PDF, 282 Kb**

## Details

### Authors and affiliations

Taekyun Kim

*Department of Mathematics, College of Science, Tianjin Polytechnic University
*

*Tianjin 300160, China, and*

Department of Mathematics, Kwangwoon University

Seoul 139-701, South Korea

Department of Mathematics, Kwangwoon University

Seoul 139-701, South Korea

Dae San Kim

*Department of Mathematics, Sogang University
Seoul 121-742, South Korea*

Toufik Mansour

*Department of Mathematics, University of Haifa
3498838 Haifa, Israel*

Gwan-Woo Jang

*Department of Mathematics, Kwangwoon University
Seoul 139-701, South Korea
*

### Abstract

We study the Fourier series of functions related to sum of products of Bernoulli polynomials and either Euler or Genocchi polynomials. As consequences, several new identities for the Bernoulli, Euler, and Genocchi functions and numbers are derived.

### Keywords

- Fourier series
- Bernoulli polynomials
- Euler polynomials
- Genocchi polynomials

### 2010 Mathematics Subject Classification

- 11B68
- 42A16

### References

- Agarwal, R. P., Kim, D. S., Kim, T., & Kwon, J. (2017) Sums of finite products of Bernoulli functions, Adv. Difference Equ., 2017, Paper No. 237, 15 pp.
- Apostol, T. M. (1950) Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J., 17, 147–157.
- Bayad, A. (2011) Special values of Lerch zeta function and their Fourier expansions, Adv. Stud. Contemp. Math. (Kyungshang), 21, 1, 1–4.
- Carlitz, L. (1953) The reciprocity theorem for Dedekind sums, Pacific J. Math., 3, 523–527.
- Carlitz, L. (1968) Some unusual congruences for the Bernoulli and Genocchi numbers, Duke Math. J., 35, 563–566.
- Ding, D., & Yang, J. (2010) Some identities related to the Apostol–Euler and Apostol– Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 20, 1, 7–21.
- Dolgy, D. V., Jang, G.-W., & Kim, T. (2017) Note on multiple
*q*-zeta functions, Adv. Stud. Contemp. Math. (Kyungshang), 27, 1, 147–154. - Dzafarli, G. M. (1964) Fourier series for functions of the space Lq in terms of a multiplicative system of functions, Azerbaidzan. Gos. Univ. Ucen. Zap. Ser. Fiz.-Mat. i Him. Nauk, 1964, 4, 11–16 (in Russian).
- Gandhi, J. M. (1960) A new formula for Genocchi numbers, Math. Student, 28, 83–85.
- Goes, G., & Goes, S. (1970) Sequences of bounded variation and sequences of Fourier coefficients, I. Math. Z., 118, 93–102.
- Guven, A., & Israfilov, D. M. (2009) Approximation by means of Fourier trigonometric series in weighted Orlicz spaces, Adv. Stud. Contemp. Math. (Kyungshang), 19, 2, 283–295.
- Jang, G.-W., Kim, D. S., Kim, T., & Mansour, T. (2017) Fourier series of functions related to Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 27, 1, 49–62.
- Kim, D. S., & Kim, T. (2012) Bernoulli basis and the product of several Bernoulli polynomials, Int. J. Math. Sci., 2012, Art. ID 463659.
- Kim, T., Kim, D. S., Jang, G.-W., & Park, J.-W. (2017) Fourier series of functions related to ordered Bell polynomials, Util. Math., 104, 67–81.
- Kim, T. (2010) Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 20, 1, 23–28.
- Kim, T. (2008) Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal., 2008, Art. ID 581582, 11 pp.
- Kim, T. (2008) Note on the Euler numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 17, 2, 131–136.
- Kim, T., Choi, J., & Kim, Y. H. (2012) A note on the values of Euler zeta functions at positive integers, Adv. Stud. Contemp. Math. (Kyungshang), 22, 1, 27–34.
- Kim, T., Kim, D. S., Dolgy, D. V., &Rim, S.-H. (2013) Some identities on the Euler numbers arising from Euler basis polynomials, ARS Combinatoria, 109, 433–446.
- Kim, T., Kim, D. S., Rim, S.-H., & Dolgy, D. V. (2017) Fourier series of higher-order Bernoulli functions and their applications, J. Inequal. Appl., 2017, Paper No. 8, 7 pp.
- Kim, T., Kim, D. S., Dolgy, D. V., & Park, J.-W. (2017) Fourier series of sums of products of ordered Bell and poly-Bernoulli functions, J. Inequal. Appl., 2017, Paper No. 84, 17 pp.
- Kim, T., Kim, D. S., Jang, L. C., & Jang, G.-W. (2017) Sums of finite products of Genocchi functions, Adv. Difference Equ., 2017, Paper No. 268, 17 pp.
- Shiratani, K. (1964) On some relations between Bernoulli numbers and class numbers of cyclotomic fields, Mem. Fac. Sci. Kyushu Univ. Ser. A, 18, 127–135.
- Watari, C. (1964) Multipliers for Walsh-Fourier series, Tohoku Math. J., 2, 16, 239–251.
- Yadav, B. H. (1964) Absolute convergence of Fourier series, Thesis (Ph.D.), Maharaja Sayajirao University of Baroda, India.
- Zill, D. G., & Cullen, M. R. (2006) Advanced Engineering Mathematics, Jones and Bartlett Publishers.

## Related papers

## Cite this paper

APAKim, T., Kim, D. S., Mansour, T. & Jang, G.-W. (2018). Fourier series of sums of products of Bernoulli and Euler/Genocchi functions. Notes on Number Theory and Discrete Mathematics, 24(1), 76-93, doi: 10.7546/nntdm.2018.24.1.76-93.

ChicagoKim, Taekyun, Dae San Kim, Toufik Mansour and Gwan-Woo Jang. “Fourier Series of Sums of Products of Bernoulli and Euler/Genocchi Functions.” Notes on Number Theory and Discrete Mathematics 24, no. 1 (2018): 76-93, doi: 10.7546/nntdm.2018.24.1.76-93.

MLAKim, Taekyun, Dae San Kim, Toufik Mansour and Gwan-Woo Jang. “Fourier Series of Sums of Products of Bernoulli and Euler/Genocchi Functions.” Notes on Number Theory and Discrete Mathematics 24.1 (2018): 76-93. Print, doi: 10.7546/nntdm.2018.24.1.76-93.