Identities of the degenerate Daehee numbers with the Bernoulli numbers of the second kind arising from nonlinear differential equation

Volume 10, Issue 12, pp 6219--6228 http://dx.doi.org/10.22436/jnsa.010.12.08

Publication Date: December 06, 2017 Submission Date: September 24, 2017

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)

• 2125 Downloads
• 4089 Views

Authors

Sung-Soo Pyo - Department of Mathematics Education, Silla University, Busan, Rep. of Korea. Taekyun Kim - Department of Mathematics, Kwangwoon University, Seoul, Rep. of Korea. Seog-Hoon Rim - Department of Mathematics Education, Kyungpook National University, Taegu, Rep. of Korea.

Abstract

In [T. Kim, D. S. Kim, H. I. Kwon, J. J. Seo, Glob. J. Pure Appl. Math., \({\bf 12}\) (2016), 1893--1901], Kim et al. presented some identities for the Bernoulli numbers of the second kind using differential equation. Here we use this differential equation in a different way. In this paper, we deduce some identities of the degenerate Daehee numbers with the Bernoulli numbers of the second kind of order \(r\).

Share and Cite

Keywords

• Degenerate Daehee numbers
• Bernoulli numbers of the second kind
• nonlinear differential equation

MSC

•  05A19
•  11B37
•  11B83
•  34A34

References

• [1] L. Carlitz, Extended Stirling and exponential numbers, Duke Math. J., 32 (1965), 205–224.
  • MathSciNet
  • Google Scholar


• [2] L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers , Utilitas Math., 15 (1979), 51–88.
  • MathSciNet
  • Google Scholar
  • MATH


• [3] D. V. Dolgy, T. Kim, L.-C. Jang, H.-I. Kwon, J. J. Seo, Some identities of symmetry for the degenerate q-Bernoulli polynomials under symmetry group of degree n, Glob. J. Pure Appl. Math., 12 (2016), 4385–4394.
  • Google Scholar


• [4] B. S. El-Desouky, A. Mustafa, New results on higher-order Daehee and Bernoulli numbers and polynomials, Adv. Difference Equ., 2016 (2016), 21 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [5] B. S. El-Desouky, A. Mustafa, F. M. Abdel-Moneim, Multiparameter Higher Order Daehee and Bernoulli Numbers and Polynomials, Applied Math., 2017 (2017), 775–785.
  • View Article
  • Google Scholar


• [6] G.-W. Jang, T. Kim, Revisit of identities of Daehee numbers arising from nonlinear differential equations , Proc. Jangjeon Math. Soc., 20 (2017), 163–177.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] G.-W. Jang, T. Kim, Some identities of ordered Bell numbers arising from differential equation, Adv. Stud. Contemp. Math., 27 (2017), 385–397.
  • Google Scholar
  • MATH


• [8] G.-W. Jang, J. Kwon, J. G. Lee , Some identities of degenerate Daehee numbers arising from nonlinear differential equation, Adv. Difference Equ., 2017 (2017), 10 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [9] T. Kim, Identities involving Frobenius-Euler polynomials arising from non-linear differential equations, J. Number Theory, 132 (2012), 2854–2865.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] T. Kim, On the degenerate Cauchy numbers and polynomials, Proc. Jangjeon Math. Soc., 18 (2015), 307–312.
  • MathSciNet
  • Google Scholar
  • MATH


• [11] T. Kim, Degenerate Cauchy numbers and polynomials of the second kind, Adv. Stud. Contemp. Math., 27 (2017), 441– 449.
  • View Article
  • Google Scholar
  • MATH


• [12] D. S. Kim, T. Kim, Some identities for Bernoulli numbers of the second kind arising from a non-linear differential equation, Bull. Korean Math. Soc., 52 (2015), 2001–2010.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] T. Kim, D. S. Kim, A note on nonlinear Changhee differential equations, Russ. J. Math. Phys., 23 (2016), 88–92.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] D. S. Kim, T. Kim , A note on degenerate Eulerian numbers and polynomials, Adv. Stud. Contemp. Math., 27 (2017), 431–440.
  • View Article
  • Google Scholar
  • MATH


• [15] D. S. Kim, T. Kim, On degenerate Bell numbers and polynomials, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math., 111 (2017), 435–446.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] D. S. Kim, T. Kim, D. V. Dolgy , A note on degenerate Bernoulli numbers and polynomials associated with p-adic invariant integral on \(\mathbb{Z}_p\), Appl. Math. Comput., 259 (2015), 198–204.
  • View Article
  • MathSciNet
  • Google Scholar


• [17] D. S. Kim, T. Kim, G. -W. Jang, Some identities of partially degenerate Touchard polynomials arising from differential equations, Adv. Stud. Contemp. Math., 27 (2017), 243–251.
  • View Article
  • Google Scholar
  • MATH


• [18] T. Kim, D. S. Kim, K.-W. Hwang, J. J. Seo, Some identities of Laguerre polynomials arising from differential equations , Adv. Difference Equ., 2016 (2016), 9 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] T. Kim, D. S. Kim, H. I. Kwon, J. J. Seo, Higher-order Cauchy Numbers and Polynomials, Appl. Math. Sci., 9 (2015), 1989–2004.
  • Google Scholar


• [20] T. Kim, D. S. Kim, H. I. Kwon, J. J. Seo, Differential equations arising from the generating function of general modified degenerate Euler numbers, Adv. Difference Equ., 2016 (2016), 7 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [21] T. Kim, D. S. Kim, H. I. Kwon, J. J. Seo, Revisit nonlinear differential equations associated with Bernoulli numbers of the second kind, Glob. J. Pure Appl. Math., 12 (2016), 1893–1901.
  • Google Scholar


• [22] T. Kim, D. S. Kim, T. Mansour, J. J. Seo, Linear differential equations for families of polynomials, J. Inequal. Appl., 2016 (2016), 8 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [23] J.-W. Park, J. Kwon, A note on the Degenerate High order Daehee polynomials, Appl. Math. Sic., 9 (2015), 4635–4642.
  • View Article
  • Google Scholar


• [24] N. L. Wang, H. Li, Some identities on the Higher-order Daehee and Changhee Numbers , Pure Appl. Math. J., 4 (2015), 33–37.
  • Google Scholar