Simplification of Coefficients in Differential Equations Associated with Higher Order Frobenius-Euler Numbers

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Abstract

In the paper, the authors apply Faà di Bruno formula, some properties of the Bell polynomials of the second kind, the inversion formulas of binomial numbers and the Stirling numbers of the first and the second kind, to significantly simplify coefficients in two families of ordinary differential equations associated with the higher order Frobenius–Euler numbers.

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  • [1] COMTET L.: : Advanced Combinatorics: The Art of Finite and Infinite Expansions Revised and Enlarged Edition D. Reidel Publishing Co. Dordrecht and Boston 1974. https://doi.org/10.1007/978-94-010-2196-8.

  • [2] GUO B.-N.—QI F.: : Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind J. Comput. Appl. Math. 272 (2014) 251–257; http://dx.doi.org/10.1016/j.cam.2014.05.018.

  • [3] ______ Some identities and an explicit formula for Bernoulli and Stirling numbers J. Comput. Appl. Math. 255 (2014) 568–579; http://dx.doi.org/10.1016/j.cam.2013.06.020.

  • [4] KIM D. S.—KIM T.—PARK J.-W.—SEO J.-J.: : Differential equations associated with higher-order Frobenius-Euler numbers Glob. J. Pure Appl. Math. 12 (2016) no. 3 2537–2547.

  • [5] KIM T.—KIM D. S.—JANG L.-C.—KWON H. I.: : Differential equations associated with higher-order Frobenius–Euler numbers revisited Differ. Equ. Dyn. Syst. (2019) (to appear); https://doi.org/10.1007/s12591-017-0380-8.

  • [6] QI F.: A simple form for coefficients in a family of nonlinear ordinary differential equations Adv. Appl. Math. Sci. 17 (2018) no. 8 555–561.

  • [7] ______ A simple form for coefficients in a family of ordinary differential equations related to the generating function of the Legendre polynomials Adv. Appl. Math. Sci. 17 (11) (2018) no. 11 693–700.

  • [8] ______ Notes on several families of differential equations related to the generating function for the Bernoulli numbers of the second kind Turkish J. Anal. Number Theory 6 (2018) no. 2 40–42; https://doi.org/10.12691/tjant-6-2-1.

  • [9] ______ Simple forms for coefficients in two families of ordinary differential equations Glob. J. Math. Anal. 6 (2018) no. 1 7–9. https://doi.org/10.14419/gjma.v6i1.9778.

  • [10] ______ Simplification of coefficients in two families of nonlinear ordinary differential equations Turkish J. Anal. Number Theory 6 (2018) no. 6 116–119; https://doi.org/10.12691/tjant-6-4-2.

  • [11] ______ Simplifying coefficients in a family of nonlinear ordinary differential equations Acta Comment. Univ. Tartu Math. 22 (2018) no. 2 293–297; https://doi.org/10.12697/ACUTM.2018.22.24.

  • [12] ______ Simplifying coefficients in a family of ordinary differential equations related to the generating function of the Laguerre polynomials Appl. Appl. Math. 13 (2018) no. 2 750–755.

  • [13] QI F.: Simplifying coefficients in differential equations related to generating functions of reverse Bessel and partially degenerate Bell polynomials Bol. Soc. Paran. Mat. 39(2021) no. 4 (to appear); http://dx.doi.org/10.5269/bspm.41758.

  • [14] QI F.—GUO B.-N.: A diagonal recurrence relation for the Stirling numbers of the first kind Appl. Anal. Discrete Math. 12 (2018) no. 1 153–165; https://doi.org/10.2298/AADM170405004Q.

  • [15] ______ Explicit formulas and recurrence relations for higher order Eulerian polynomials Indag. Math. 28 (2017) no. 4 884–891 https://doi.org/10.1016/j.indag.2017.06.010.

  • [16] ______ Some properties of the Hermite polynomials. In: Advances in Special Functions and Analysis of Differential Equations (Praveen Agarwal Ravi P. Agarwal and Michael Ruzhansky eds.) CRC Press Taylor & Francis Group 2019 (to appear).

  • [17] ______ Viewing some ordinary differential equations from the angle of derivative polynomials Iran. J. Math. Sci. Inform. 14 (2019) no. 2 (to appear).

  • [18] QI F.—LIM D.—GUO B.-N.: : Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113 (2019) no. 1 1–9; http://dx.doi.org/10.1007/s13398-017-0427-2.

  • [19] ______ Some identities related to Eulerian polynomials and involving the Stirling numbers Appl. Anal. Discrete Math. 12 (2018) no. 2. 467–480; https://doi.org/10.2298/AADM171008014Q.

  • [20] QI F.—LIM D.—LIU A.-Q.: : Explicit expressions related to degenerate Cauchy numbers and their generating function In: Proceeding on Mathematical Modelling Applied Analysis and Computation In: Springer Proceedings in Mathematics and Statistics 2019 (to appear); https://www.springer.com/series/10533.

  • [21] QI F.—NIU D.-W.—GUO B.-N.: : Simplification of coefficients in differential equations associated with higher order Frobenius-Euler numbers Preprints 2017 2017080017 7 pp; http://dx.doi.org/10.20944/preprints201708.0017.v1.

  • [22] ______ Some identities for a sequence of unnamed polynomials connected with the Bell polynomials Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 112 (2018) (to appear); https://doi.org/10.1007/s13398-018-0494-z.

  • [23] QI F.—WANG J.-L.—GUO B.-N.: : Notes on a family of inhomogeneous linear ordinary differential equations Adv. Appl. Math. Sci. 17 (2018) no. 4 361–368.

  • [24] ______ Simplifying and finding ordinary differential equations in terms of the Stirling numbers Korean J. Math. 26 (2018) no. 4 675–681; https://doi.org/10.11568/kjm.2018.26.4.675.

  • [25] ______ Simplifying differential equations concerning degenerate Bernoulli and Euler numbers Trans. A. Razmadze Math. Inst. 172 (2018) no. 1 90–94; http://dx.doi.org/10.1016/j.trmi.2017.08.001.

  • [26] QI F.—ZHAO J.-L.: : Some properties of the Bernoulli numbers of the second kind and their generating function Bull. Korean Math. Soc. 55 (2018) no. 6 1909–1920; https://doi.org/10.4134/BKMS.b180039.

  • [27] QUAINTANCE J.—GOULD H. W.: : Combinatorial Identities for Stirling Numbers. The unpublished notes of H. W. Gould. With a foreword by George E. Andrews. World Scientific Publishing Co. Pte. Ltd. Singapore 2016.

  • [28] ZHAO J.-L.—WANG J.-L.—QI F.: : Derivative polynomials of a function related to the Apostol–Euler and Frobenius–Euler numbers J. Nonlinear Sci. Appl. 10 (2017) no. 4; 1345–1349 http://dx.doi.org/10.22436/jnsa.010.04.06.

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