Volume 10, Issue 5, pp 2384--2401
Publication Date: 2017-05-24
Taekyun Kim - 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.
Dmitry V. Dolgy - Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea.
Jin-Woo Park - Department of Mathematics Education, Daegu University, Gyeongsan-si, Gyeongsangbuk-do, 712-714, Republic of Korea.
In this paper, we consider three types of sums of products of poly-Bernoulli functions and derive Fourier series expansions of them. In addition, we express those three types of functions in terms of Bernoulli functions.
Fourier series, Bernoulli polynomial, poly-Bernoulli polynomial, poly-Bernoulli function.
 T. Arakawa, M. Kaneko, On poly-Bernoulli numbers, Comment. Math. Univ. St. Paul., 48 (1999), 159–167.
 A. Bayad, Y. Hamahata, Multiple polylogarithms and multi-poly-Bernoulli polynomials, Funct. Approx. Comment. Math., 46 (2012), 45–61.
 D. V. Dolgy, D. S. Kim, T. Kim, T. Mansour, Degenerate poly-Bernoulli polynomials of the second kind, J. Comput. Anal. Appl., 21 (2016), 954–966.
 G. V. Dunne, C. Schubert, Bernoulli number identities from quantum field theory and topological string theory, Commun. Number Theory Phys., 7 (2013), 225–249.
 C. Faber, R. Pandharipande, Hodge integrals and Gromov-Witten theory, Invent. Math., 139 (2000), 173–199.
 I. M. Gessel, On Miki’s identity for Bernoulli numbers, J. Number Theory, 110 (2005), 75–82.
 M. Kaneko, Poly-Bernoulli numbers, J. Théor. Nombres Bordeaux, 9 (1997), 221–228.
 D. S. Kim, D. V. Dolgy, T. Kim, S.-H. Rim, Some formulae for the product of two Bernoulli and Euler polynomials, Abstr. Appl. Anal., 2012 (2012), 15 pages.
 D. S. Kim, T. Kim, Bernoulli basis and the product of several Bernoulli polynomials, Int. J. Math. Math. Sci., 2012 (2012), 12 pages.
 D. S. Kim, T. Kim, Some identities of higher order Euler polynomials arising from Euler basis, Integral Transforms Spec. Funct., 24 (2013), 734–738.
 D. S. Kim, T. Kim, A note on degenerate poly-Bernoulli numbers and polynomials, Adv. Difference Equ., 2015 (2015), 8 pages.
 D. S. Kim, T. Kim, A note on poly-Bernoulli and higher-order poly-Bernoulli polynomials, Russ. J. Math. Phys., 22 (2015), 26–33.
 D. S. Kim, T. Kim, Higher-order Bernoulli and poly-Bernoulli mixed type polynomials, Georgian Math. J., 22 (2015), 265–272.
 D. S. Kim, T. Kim, H. I. Kwon, T. Mansour, Degenerate poly-Bernoulli polynomials with umbral calculus viewpoint, J. Inequal. Appl., 2015 (2015), 13 pages.
 D. S. Kim, T. Kim, T. Mansour, J.-J. Seo, Fully degenerate poly-Bernoulli polynomials with a q parameter, Filomat, 30 (2016), 1029–1035.
 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.
 T. Kim, D. S. Kim, J.-J. Seo, Fully degenerate poly-Bernoulli numbers and polynomials, Open Math., 14 (2016), 545–556.
 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, Istv´an F´ary and Robert Gulliver, W. H. Freeman and Co., San Francisco, (1974).
 K. Shiratani, S. Yokoyama, An application of p-adic convolutions, Mem. Fac. Sci. Kyushu Univ. Ser. A, 36 (1982), 73–83.
 P. T. Young, Bernoulli and poly-Bernoulli polynomial convolutions and identities of p-adic Arakawa-Kaneko zeta functions, J. Number Theory, 12 (2016), 1295–1309.
 D. G. Zill, M. R. Cullen, Advanced engineering mathematics, second edition, Jones & Bartlett Learning, Massachusetts, (2000).