[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.b160863

FOURIER SERIES OF HIGHER-ORDER EULER FUNCTIONS AND THEIR APPLICATIONS

Kim, Dae San (Department of Mathematics Sogang University)
Kim, Taekyun (Department of Mathematics Kwangwoon University)

Publication Information

Bulletin of the Korean Mathematical Society / v.55, no.1, 2018 , pp. 107-114 More about this Journal

Abstract

In this paper, we give some identities for the higher-order Euler functions arising from the Fourier series of them. In addition, we investigate some formulae related to Bernoulli functions which are derived from our identities.

Keywords

Fourier series; Euler polynomials; Euler functions;

Citations & Related Records

Reference

1	J. Higgins, Double series for the Bernoulli and Euler numbers, J. London Math. Soc. 2 (1970), 722-726. DOI
2	D. S. Kim and T. Kim, Generalized Boole numbers and polynomials, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM. 110 (2016), no. 2, 823-839.
3	D. S. Kim, T. Kim, H.-I. Kwon, and T. Mansour, Nonlinear differential equation for Korobov numbers, Adv. Stud. Contemp. Math. (Kyungshang) 26 (2016), no. 4, 733-740.
4	D. S. Kim, T. Kim, H.-I. Kwon, and T. Mansour, Barnes-type Boole polynomials, Contrib. Discrete Math. 11 (2016), no. 1, 7-15.
5	T. Kim, Note on the Euler numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 17 (2008), no. 2, 131-136.
6	T. Kim, Euler numbers and polynomials associated with zeta functions, Abstr. Apol. Anal. 2008 (2008), Art. ID 581582, 11 pp.
7	T. Kim, Some identities for the Bernoulli the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 1, 23-28.
8	T. Kim, J. Choi, and Y. H. Kim, A note on the values of Euler zeta functions at positive integers, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 1, 27-34.
9	T. Kim and D. S. Kim, On $\lambda$ -Bell Polynomials associated with umbral calculus, Russ. J. Math. Phys. 24 (2017), no. 1, 1-10. DOI
10	T. Kim and D. S. Kim, A note on nonlinear Changhee differential equations, Russ. J. Math. Phys. 23 (2016), no. 1, 88-92. DOI
11	F. R. Olson, Some determinants involving Bernoulli and Euler numbers of higher order, Pacific J. Math. 5 (1955), 259-268. DOI
12	T. Kim and D. S. Kim, Identities involving degenerate Euler numbers and polynomials arising from non-linear differential equations, J. Nonlinear Sci. Appl. 9 (2016), no. 5, 2086-2098. DOI
13	H. I. Kwon, T. Kim, and J. J. Seo, Some new identities of symmetry for modified degenerate Euler polynomials, Proc. Jangjeon Math. Soc. 19 (2016), no. 2, 237-242.
14	D. H. Lehmer, A new approach to Bernoulli polynomials, Amer. Math. Monthly. 95 (1988), no. 10, 905-911. DOI
15	L. C. Washington, Introduction to Cyclotomic Fields, Second edition. Graduate Text in Mathematics 83, Springer-Verlag. New York, 1997.
16	A. Sharma, q-Bernoulli and Euler numbers of higher order, Duke Math. J. 25 (1958), 343-353. DOI
17	L. Carlitz, The multiplication formulas for the Bernoulli and Euler polynomials, Math. Mag. 27 (1953), 59-64. DOI
18	M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970.
19	E. M. Beesley, An integral representation for the Euler numbers, Amer. Math. Monthly 76 (1969), 389-391. DOI
20	L. Carlitz, Some formulas for the Bernoulli and Euler polynomials, Math. Nachr. 25 (1963), 223-231. DOI