Derivative polynomials of a function related to the Apostol-Euler and Frobenius-Euler numbers

Volume 10, Issue 4, pp 1345--1349 http://dx.doi.org/10.22436/jnsa.010.04.06

Publication Date: April 20, 2017 Submission Date: January 22, 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)

• 2430 Downloads
• 7161 Views

Authors

Jiao-Lian Zhao - Department of Mathematics and Physics, Weinan Normal University, Weinan City, Shaanxi Province, 714009, China. Jing-Lin Wang - Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China. Feng Qi - Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China. - Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China.

Abstract

In the paper, the authors find a simple and significant expression in terms of the Stirling numbers for derivative polynomials of a function with a parameter related to the higher order Apostol-Euler numbers and to the higher order Frobenius-Euler numbers. Moreover, the authors also present a common solution to a sequence of nonlinear ordinary differential equations.

Share and Cite

Keywords

• Derivative polynomial
• Stirling number
• nonlinear ordinary differential equation
• solution.

MSC

•  11B68
•  11B73
•  34A34

References

• [1] B.-N. Guo, F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math., 272 (2014), 251–257.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] B.-N. Guo, F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math., 255 (2014), 568–579.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] M. E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly, 102 (1995), 23–30.
  • View Article
  • MathSciNet
  • Google Scholar


• [4] T. Kim, G.-W. Jang, J. J. Seo, Revisit of identities for Apostol-Euler and Frobenius-Euler numbers arising from differential equation, J. Nonlinear Sci. Appl., 10 (2017), 186–191.
  • Google Scholar


• [5] T. Kim, D. S. Kim, Some identities of Eulerian polynomials arising from nonlinear differential equations, Iran. J. Sci. Technol. Trans. A Sci., 2016 (2016 ), 6 pages.


• [6] T. Kim, D. S. Kim, Differential equations associated with Catalan–Daehee numbers and their applications, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 111 (2017), 1–11.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] T. Kim, D. S. Kim, L.-C. Jang, H. I. Kwon, On differential equations associated with squared Hermite polynomials, J. Comput. Anal. Appl., 23 (2017), 1252–1264.
  • MathSciNet
  • Google Scholar


• [8] T. Kim, D. S. Kim, J.-J. Seo, D. V. Dolgy, Some identities of Chebyshev polynomials arising from non-linear differential equations, J. Comput. Anal. Appl., 23 (2017), 820–832.
  • MathSciNet
  • Google Scholar


• [9] F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput., 268 (2015), 844–858.
  • View Article
  • MathSciNet
  • Google Scholar


• [10] F. Qi, Explicit formulas for the convolved Fibonacci numbers, ResearchGate Working Paper, (2016), 9 pages.
  • Google Scholar


• [11] F. Qi, B.-N. Guo, Explicit formulas and nonlinear ODEs of generating functions for Eulerian polynomials, ResearchGate Working Paper, (2016), 5 pages.
  • Google Scholar


• [12] F. Qi, B.-N. Guo, Some properties of a solution to a family of inhomogeneous linear ordinary differential equations, Preprints, 2016 (2016), 11 pages.
  • Google Scholar


• [13] F. Qi, B.-N. Guo, Some properties of the Hermite polynomials and their squares and generating functions, Preprints, 2016 (2016), 14 pages.
  • View Article
  • Google Scholar


• [14] F. Qi, B.-N. Guo, Viewing some nonlinear ODEs and their solutions from the angle of derivative polynomials, ResearchGate Working Paper, (2016), 10 pages.
  • Google Scholar


• [15] F. Qi, B.-N. Guo, Viewing some ordinary differential equations from the angle of derivative polynomials, Preprints, 2016 (2016), 12 pages.
  • View Article
  • Google Scholar


• [16] F. Qi, J.-L. Zhao, The Bell polynomials and a sequence of polynomials applied to differential equations, Preprints, 2016 (2016 ), 8 pages.
  • Google Scholar


• [17] F. Qi, J.-L. Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, J. Differ. Equ. Appl., (2017), in press.
  • Google Scholar


• [18] C.-F. Wei, B.-N. Guo, Complete monotonicity of functions connected with the exponential function and derivatives, Abstr. Appl. Anal., 2014 (2014), 5 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [19] A.-M. Xu, Z.-D. Cen, Some identities involving exponential functions and Stirling numbers and applications, J. Comput. Appl. Math.,, 260 (2014), 201–207.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] A.-M. Xu, Z.-D. Cen, Closed formulas for computing higher-order derivatives of functions involving exponential functions, Appl. Math. Comput., 270 (2015), 136–141.
  • View Article
  • MathSciNet
  • Google Scholar


• [21] J.-L. Zhao, J.-L. Wang, F. Qi, Derivative polynomials of a function related to the Apostol–Euler and Frobenius–Euler numbers, ResearchGate Working Paper, (2017), 5 pages.
  • View Article
  • MathSciNet
  • Google Scholar