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

Zhao, Jiao-Lian¹ ; Wang, Jing-Lin² ; Qi, Feng³

¹ Department of Mathematics and Physics, Weinan Normal University, Weinan City, Shaanxi Province, 714009, China
² Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China
³ 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

Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 4, p. 1345-1349.

Résumé

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.

Détail

DOI : 10.22436/jnsa.010.04.06

Classification : 11B68, 11B73, 34A34
Keywords: Derivative polynomial, Stirling number, nonlinear ordinary differential equation, solution.

Bibliographie

[1] Guo, B.-N.; Qi, F. Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math., Volume 272 (2014), pp. 251-257 | DOI | Zbl

[2] Guo, B.-N.; Qi, F. Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math., Volume 255 (2014), pp. 568-579 | Zbl | DOI

[3] Hoffman, M. E. Derivative polynomials for tangent and secant, Amer. Math. Monthly, Volume 102 (1995), pp. 23-30 | DOI

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

[5] Kim, T.; Kim, D. S. Some identities of Eulerian polynomials arising from nonlinear differential equations, Iran. J. Sci. Technol. Trans. A Sci., Volume 2016 (2016 ), pp. 1-6

[6] Kim, T.; Kim, D. S. Differential equations associated with Catalan–Daehee numbers and their applications, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, Volume 111 (2017), pp. 1-11 | Zbl | DOI

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

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

[9] Qi, F. Derivatives of tangent function and tangent numbers, Appl. Math. Comput., Volume 268 (2015), pp. 844-858 | DOI

[10] Qi, F. Explicit formulas for the convolved Fibonacci numbers, ResearchGate Working Paper (2016), pp. 1-9

[11] Qi, F.; Guo, B.-N. Explicit formulas and nonlinear ODEs of generating functions for Eulerian polynomials, ResearchGate Working Paper (2016), pp. 1-5

[12] Qi, F.; Guo, B.-N. Some properties of a solution to a family of inhomogeneous linear ordinary differential equations, Preprints, Volume 2016 (2016), pp. 1-11

[13] Qi, F.; Guo, B.-N. Some properties of the Hermite polynomials and their squares and generating functions, Preprints, Volume 2016 (2016), pp. 1-14

[14] Qi, F.; Guo, B.-N. Viewing some nonlinear ODEs and their solutions from the angle of derivative polynomials, ResearchGate Working Paper (2016), pp. 1-10

[15] Qi, F.; Guo, B.-N. Viewing some ordinary differential equations from the angle of derivative polynomials, Preprints, Volume 2016 (2016), pp. 1-12

[16] Qi, F.; Zhao, J.-L. The Bell polynomials and a sequence of polynomials applied to differential equations, Preprints, Volume 2016 (2016 ), pp. 1-8

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

[18] Wei, C.-F.; Guo, B.-N. Complete monotonicity of functions connected with the exponential function and derivatives, Abstr. Appl. Anal., Volume 2014 (2014), pp. 1-5

[19] Xu, A.-M.; Cen, Z.-D. Some identities involving exponential functions and Stirling numbers and applications, J. Comput. Appl. Math.,, Volume 260 (2014), pp. 201-207 | DOI | Zbl

[20] Xu, A.-M.; Cen, Z.-D. Closed formulas for computing higher-order derivatives of functions involving exponential functions, Appl. Math. Comput., Volume 270 (2015), pp. 136-141 | DOI

[21] Zhao, J.-L.; Wang, J.-L.; Qi, F. Derivative polynomials of a function related to the Apostol–Euler and Frobenius–Euler numbers, ResearchGate Working Paper (2017), pp. 1-5