Fourier series of sums of products of Genocchi functions and their applications

Volume 10, Issue 4, pp 1683--1694 http://dx.doi.org/10.22436/jnsa.010.04.31

Publication Date: April 20, 2017 Submission Date: December 19, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1979 Downloads
• 3327 Views

Authors

Taekyun Kim - Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin 300160, China. - Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea. Dae San Kim - Department of Mathematics, Sogang University, Seoul, 121-742, Republic of Korea. Gwan-Woo Jang - Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea. Jongkyum Kwon - Department of Mathematics Education and RINS, Gyeongsang National University, Jinju, Gyeongsangnamdo, 52828, Republic of Korea.

Abstract

Recently, Luo introduced Fourier expansions of Apostol-Bernoulli, Apostol-Euler and Genocchi polynomials and investigated some interesting identities and properties of these polynomials by using Fourier series. In this paper, we consider three types of functions given by sums of products of Genocchi functions and derive their Fourier series expansions. In addition, we will express each of them in terms of Bernoulli functions.

Share and Cite

Keywords

• Fourier series
• Genocchi functions
• Genocchi polynomials.

MSC

•  11B83
•  42A16

References

• [1] A. Bayad, T. Kim, Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 20 (2010), 247–253.
  • MathSciNet
  • Google Scholar
  • MATH


• [2] A. Bayad, T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russ. J. Math. Phys., 18 (2011), 133–143.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] A. Bayad, T. Kim, Higher recurrences for Apostol-Bernoulli-Euler numbers, Russ. J. Math. Phys., 19 (2012), 1–10.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] G. V. Dunne, C. Schubert, Bernoulli number identities from quantum field theory and topological string theory, Commun. Number Theory Phys., 7 (2013), 225–249.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] C. Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math., 139 (2000), 173–199.
  • View Article
  • MathSciNet
  • Google Scholar


• [6] T. Kim, A note on the q-Genocchi numbers and polynomials, J. Inequal. Appl., 2007 (2007), 8 pages,.
  • View Article
  • Google Scholar


• [7] T. Kim, On the q-extension of Euler and Genocchi numbers, J. Math. Anal. Appl., 326 (2007), 1458–1465.
  • View Article
  • MathSciNet
  • Google Scholar


• [8] T. Kim, Euler numbers and polynomials associated with zeta functions, Abstr. Appl. Anal., 2008 (2008), 11 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [9] T. Kim, Note on the Euler numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 17 (2008), 131–136.


• [10] T. Kim, On the multiple q-Genocchi and Euler numbers, Russ. J. Math. Phys., 15 (2008), 481–486.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang), 20 (2010), 23–28.
  • MathSciNet
  • Google Scholar
  • MATH


• [12] D. S. Kim, T. Kim, Bernoulli basis and the product of several Bernoulli polynomials, Int. J. Math. Math. Sci., 2012 (2012), 12 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [13] D. S. Kim, T. Kim, Euler basis, identities, and their applications, Int. J. Math. Math. Sci., 2012 (2012), 15 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] D. S. Kim, T. Kim, A note on higher-order Bernoulli polynomials, J. Inequal. Appl., 2013 (2013), 9 pages.
  • MathSciNet


• [15] D. S. Kim, T. Kim, Some identities of higher order Euler polynomials arising from Euler basis, Integral Transforms Spec. Funct., 24 (2013), 734–738.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] D. S. Kim, T. Kim, Identities arising from higher-order Daehee polynomial bases, Open Math., 13 (2015), 196–208.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] T. Kim, D. S. Kim, S.-H. Rim, D. V. Dolgy, Fourier series of higher-order Bernoulli functions and their applications, J. Inequal. Appl., 2017 (2017), 7 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [18] Q.-M. Luo, Extensions of the Genocchi polynomials and their Fourier expansions and integral representations, Osaka J. Math., 48 (2011), 291–309.
  • MathSciNet
  • Google Scholar
  • MATH


• [19] J. E. Marsden, Elementary classical analysis, With the assistance of Michael Buchner, Amy Erickson, Adam Hausknecht, Dennis Heifetz, Janet Macrae and William Wilson, and with contributions by Paul Chernoff, István Fáry and Robert Gulliver. W. H. Freeman and Co., San Francisco (1974)
  • MathSciNet


• [20] K. Shiratani, S. Yokoyama, An application of p-adic convolutions, Mem. Fac. Sci. Kyushu Univ. Ser. A, 36 (1982), 73–83.
  • View Article
  • MathSciNet
  • Google Scholar


• [21] H. M. Srivastava, Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inf. Sci., 5 (2011), 390–414.
  • MathSciNet
  • Google Scholar


• [22] D. Zill, M. R. Cullen, Advanced engineering mathematics, Second edition, Jones and Bartlett, Massachusetts (2006)
  • Google Scholar