Derivative polynomials of a function related to the ApostolEuler and FrobeniusEuler numbers
Authors
JiaoLian Zhao
 Department of Mathematics and Physics, Weinan Normal University, Weinan City, Shaanxi Province, 714009, China.
JingLin Wang
 Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China.
Feng Qi
 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 ApostolEuler numbers and to the higher order FrobeniusEuler
numbers. Moreover, the authors also present a common solution to a sequence of nonlinear ordinary differential equations.
Keywords
 Derivative polynomial
 Stirling number
 nonlinear ordinary differential equation
 solution.
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.

[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.

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

[4]
T. Kim, G.W. Jang, J. J. Seo, Revisit of identities for ApostolEuler and FrobeniusEuler numbers arising from differential equation, J. Nonlinear Sci. Appl., 10 (2017), 186–191.

[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.

[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.

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

[9]
F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput., 268 (2015), 844–858.

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

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

[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.

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

[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.

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

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

[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.

[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.

[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.

[20]
A.M. Xu, Z.D. Cen, Closed formulas for computing higherorder derivatives of functions involving exponential functions, Appl. Math. Comput., 270 (2015), 136–141.

[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.