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Original Article

Open Access Peer-reviewed

Feng Qi^{ }, Bai-Ni Guo

Received June 29, 2018; Revised July 30, 2018; Accepted September 17, 2018

In the paper, in terms of the Stirling numbers of the first and second kinds, by three approaches, the author derives simple, meaningful, and significant forms for coefficients in two families of nonlinear ordinary differential equations.

In [^{ 6}, Theorems 2.1 and 2.2], it was established inductively and recursively that the function

(1) |

satisfies the families of nonlinear ordinary differential functions

(2) |

and

(3) |

for , where , , , ,

(4) |

and

(5) |

for 0 < *k* < *n*.

In this paper, since

(1) the original proofs of [^{ 6}, Theorems 2.1 and 2.2] are long and tedious,

(2) the expressions in (4) and (5) are too complex to be remembered, understood, and computed easily,

we will provide three simple and standard proofs for [^{ 6}, Theorems 2.1 and 2.2] and, more importantly, derive simple, meaningful, and significant expressions for the quantities and .

Our main results can be stated as the following theorem.

**Theorem 1. ***For *,* the function ** defined by** *(1) *satisfies*

(6) |

*and*

(7) |

*where ** and ** stand for the Stirling numbers of the first and second kinds*.

In this section, we provide three proofs for Theorem 1 as follows.

*First proof*. It is well known [^{ 1}, Theorem 11.4] and [^{ 2}, p. 139, Theorem C] that the Faàdi Bruno formula can be described in terms of the Bell polynomials of the second kind by

(8) |

The identities

(9) |

and

(10) |

for and can be found in [^{ 1},p. 412] and [^{ 2}, p. 135]. Applying (8), (9), and (10) in sequence and denoting yield

Combining this with the identity

(11) |

in [^{ 24}, Theorem 2.2] results in (6).

The identity (7) follows from applying [^{ 2}, p. 213, eq. (5c)] or [^{ 34}, p. 171, Theorem 12.1], which reads that

for a collection of constants and independent of , to (6). The first proof of Theorem 1 is complete.

*Second proof*. A sequence of polynomials of order are called ^{ 5, 9, 22} the derivative polynomials of a function if and only if for . In [^{ 38}, Theorem 1.1], it was obtained that the derivative polynomials of the function can be computed by

(12) |

for and . In [^{ 38}, Theorem 1.3], it was obtained that the nonlinear differential equations

(13) |

have a common solution for and . Letting in (12) and (13) gives

and

The second proof of Theorem 1 is thus complete.

*Third proof*. By virtue of Theorem 2.1 in ^{ 3}, Theorems 3.1 and 3.2 in ^{ 37}, and Lemma 2.1 in ^{ 38}, it follows that

(14) |

and

(15) |

where are real constants, either and or and . See also ^{ 4, 35, 36}.

Taking and in (14) and (15) leads to

and

These two identities can be further rearranged as

and

The third proof of Theorem 1 is thus complete.

Finally, we list several remarks on our main results and closely related things.

*Remark 1.* Comparing (2) and (3) with (6) and (7) figures out that

and

which are simpler, more meaningful, and more significant than the expressions in (4) and (5).

*Remark 2.* In [^{ 34}, p. 118, Eq. (9.18)], it is listed that

This identity is different from (11) and

(16) |

for in [^{ 24}, Theorem 2.2].

*Remark 3*. Any one among three proofs is simpler and shorter than the one in the paper ^{ 6}.

*Remark 4*. The motivations in the papers ^{ 3, 4, 7, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 38} are same as the one in this paper.

*Remark 5*. This paper is a slightly modified version of the preprint ^{ 13}.

[1] | C. A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002. | ||

In article | PubMed | ||

[2] | L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974. | ||

In article | View Article | ||

[3] | B.-N. Guo and 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. | ||

In article | View Article | ||

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

In article | View Article | ||

[5] | M. E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly 102 (1995), no. 1, 23-30. | ||

In article | |||

[6] | G.-W. Jang and T. Kim, Some identities of ordered Bell numbers arising from differential equation, Adv. Stud. Comtemp. Math. 27 (2017), no. 3, 385-397. | ||

In article | |||

[7] | F. Qi, A simple form for coefficients in a family of nonlinear ordinary differential equations, Adv. Appl. Math. Sci. 17 (2018), no. 8, 555-561. | ||

In article | |||

[8] | F. Qi, 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 (2018), in press; ResearchGate Preprint (2017). | ||

In article | View Article | ||

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

In article | View Article | ||

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

In article | View Article | ||

[11] | F. Qi, 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. | ||

In article | |||

[12] | F. Qi, Simple forms for coefficients in two families of ordinary differential equations, Glob. J. Math. Anal. 6 (2018), no. 1, 7-9. | ||

In article | |||

[13] | F. Qi, Simplification of coefficients in two families of nonlinear ordinary differential equations, ResearchGate Preprint (2017). | ||

In article | View Article | ||

[14] | F. Qi, Simplifying coefficients in a family of nonlinear ordinary differential equations, Acta Comment. Univ. Tartu. Math. (2018), in press; ResearchGate Preprint (2017). | ||

In article | View Article | ||

[15] | F. Qi, Simplifying coefficients in a family of ordinary differential equations related to the generating function of the Laguerre polynomials, ResearchGate Preprint (2017). | ||

In article | View Article | ||

[16] | F. Qi, Simplifying coefficients in a family of ordinary differential equations related to the generating function of the Mittag-Leffler polynomials, ResearchGate Preprint (2017). | ||

In article | View Article | ||

[17] | F. Qi, Simplifying coefficients in differential equations related to generating functions of reverse Bessel and par- tially degenerate Bell polynomials, Bol. Soc. Paran. Mat. (2019), in press; ResearchGate Preprint (2017). | ||

In article | View Article | ||

[18] | F. Qi and B.-N. Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Appl. Anal. Discrete Math. 12 (2018), no. 1, 153-165. | ||

In article | |||

[19] | F. Qi and B.-N. Guo, Explicit formulas and recurrence relations for higher order Eulerian polynomials, Indag. Math. 28 (2017), no. 4, 884-891. | ||

In article | |||

[20] | F. Qi and B.-N. Guo, Explicit formulas for derangement numbers and their generating function, J. Nonlinear Funct. Anal. 2016, Article ID 45, 10 pages. | ||

In article | |||

[21] | F. Qi and B.-N. Guo, Some properties of the Hermite polynomials and their squares and generating functions, Preprints 2016, 2016110145, 14 pages. | ||

In article | View Article | ||

[22] | F. Qi and B.-N. Guo, Viewing some ordinary differential equations from the angle of derivative polynomials, Iran. J. Math. Sci. Inform. 14 (2019), no. 2, in press; Preprints 2016, 2016100043, 12 pages. | ||

In article | View Article | ||

[23] | F. Qi, D. Lim, and B.-N. Guo, 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 (2018), in press. | ||

In article | View Article | ||

[24] | F. Qi, D. Lim, and B.-N. Guo, Some identities related to Eulerian polynomials and involving the Stirling numbers, Appl. Anal. Discrete Math. 12 (2018), no. 2, in press. | ||

In article | View Article | ||

[25] | F. Qi, D.-W. Niu, and B.-N. Guo, Simplification of coefficients in differential equations associated with higher order Frobenius-Euler numbers, Preprints 2017, 2017080017, 7 pages. | ||

In article | View Article | ||

[26] | F. Qi, D.-W. Niu, and B.-N. Guo, Simplifying coefficients in differential equations associated with higher or- der Bernoulli numbers of the second kind, Preprints 2017, 2017080026, 6 pages. | ||

In article | View Article | ||

[27] | F. Qi, D.-W. Niu, and B.-N. Guo, 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), in press. | ||

In article | View Article | ||

[28] | F. Qi, X.-L. Qin, and Y.-H. Yao, The generating funtion of the Catalan numbers and lower triangular integer matrices, Preprints 2017, 2017110120, 12 pages. | ||

In article | View Article | ||

[29] | F. Qi, J.-L. Wang, and B.-N. Guo, Notes on a family of inhomogeneous linear ordinary differential equations, Adv. Appl. Math. Sci. 17 (2018), no. 4, 361-368. | ||

In article | |||

[30] | F. Qi, J.-L. Wang, and B.-N. Guo, Simplifying and finding nonlinear ordinary differential equations, ResearchGate Working Paper (2017). | ||

In article | |||

[31] | F. Qi, J.-L. Wang, and B.-N. Guo, Simplifying differential equations concerning degenerate Bernoulli and Euler numbers, Trans. A. Razmadze Math. Inst. 172 (2018), no. 1, 90-94. | ||

In article | |||

[32] | F. Qi and J.-L. Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, Bull. Korean Math. Soc. 55 (2018). | ||

In article | |||

[33] | F. Qi, Q. Zou, and B.-N. Guo, Some identities and a matrix inverse related to the Chebyshev polynomials of the second kind and the Catalan numbers, Preprints 2017, 2017030209, 25 pages. | ||

In article | View Article | ||

[34] | J. Quaintance and H. W. Gould, 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. | ||

In article | |||

[35] | C.-F. Wei and B.-N. Guo, Complete monotonicity of functions connected with the exponential function and derivatives, Abstr. Appl. Anal. 2014 (2014), Article ID 851213, 5 pages. | ||

In article | |||

[36] | A.-M. Xu and G.-D. Cen, Closed formulas for computing higher-order derivatives of functions involving exponential functions, Appl. Math. Comput. 270 (2015), 136-141. | ||

In article | View Article | ||

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

In article | View Article | ||

[38] | J.-L. Zhao, J.-L. Wang, and F. Qi, Derivative polynomials of a function related to the Apostol-Euler and Frobenius-Euler numbers, J. Nonlinear Sci. Appl. 10 (2017), no. 4, 1345-1349. | ||

In article | |||

Published with license by Science and Education Publishing, Copyright © 2018 Feng Qi and Bai-Ni Guo

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Feng Qi, Bai-Ni Guo. Simplification of Coefficients in Two Families of Nonlinear Ordinary Differential Equations. *Turkish Journal of Analysis and Number Theory*. Vol. 6, No. 4, 2018, pp 116-119. http://pubs.sciepub.com/tjant/6/4/2

Qi, Feng, and Bai-Ni Guo. "Simplification of Coefficients in Two Families of Nonlinear Ordinary Differential Equations." *Turkish Journal of Analysis and Number Theory* 6.4 (2018): 116-119.

Qi, F. , & Guo, B. (2018). Simplification of Coefficients in Two Families of Nonlinear Ordinary Differential Equations. *Turkish Journal of Analysis and Number Theory*, *6*(4), 116-119.

Qi, Feng, and Bai-Ni Guo. "Simplification of Coefficients in Two Families of Nonlinear Ordinary Differential Equations." *Turkish Journal of Analysis and Number Theory* 6, no. 4 (2018): 116-119.

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[1] | C. A. Charalambides, Enumerative Combinatorics, CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, 2002. | ||

In article | PubMed | ||

[2] | L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974. | ||

In article | View Article | ||

[3] | B.-N. Guo and 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. | ||

In article | View Article | ||

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

In article | View Article | ||

[5] | M. E. Hoffman, Derivative polynomials for tangent and secant, Amer. Math. Monthly 102 (1995), no. 1, 23-30. | ||

In article | |||

[6] | G.-W. Jang and T. Kim, Some identities of ordered Bell numbers arising from differential equation, Adv. Stud. Comtemp. Math. 27 (2017), no. 3, 385-397. | ||

In article | |||

[7] | F. Qi, A simple form for coefficients in a family of nonlinear ordinary differential equations, Adv. Appl. Math. Sci. 17 (2018), no. 8, 555-561. | ||

In article | |||

[8] | F. Qi, 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 (2018), in press; ResearchGate Preprint (2017). | ||

In article | View Article | ||

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

In article | View Article | ||

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

In article | View Article | ||

[11] | F. Qi, 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. | ||

In article | |||

[12] | F. Qi, Simple forms for coefficients in two families of ordinary differential equations, Glob. J. Math. Anal. 6 (2018), no. 1, 7-9. | ||

In article | |||

[13] | F. Qi, Simplification of coefficients in two families of nonlinear ordinary differential equations, ResearchGate Preprint (2017). | ||

In article | View Article | ||

[14] | F. Qi, Simplifying coefficients in a family of nonlinear ordinary differential equations, Acta Comment. Univ. Tartu. Math. (2018), in press; ResearchGate Preprint (2017). | ||

In article | View Article | ||

[15] | F. Qi, Simplifying coefficients in a family of ordinary differential equations related to the generating function of the Laguerre polynomials, ResearchGate Preprint (2017). | ||

In article | View Article | ||

[16] | F. Qi, Simplifying coefficients in a family of ordinary differential equations related to the generating function of the Mittag-Leffler polynomials, ResearchGate Preprint (2017). | ||

In article | View Article | ||

[17] | F. Qi, Simplifying coefficients in differential equations related to generating functions of reverse Bessel and par- tially degenerate Bell polynomials, Bol. Soc. Paran. Mat. (2019), in press; ResearchGate Preprint (2017). | ||

In article | View Article | ||

[18] | F. Qi and B.-N. Guo, A diagonal recurrence relation for the Stirling numbers of the first kind, Appl. Anal. Discrete Math. 12 (2018), no. 1, 153-165. | ||

In article | |||

[19] | F. Qi and B.-N. Guo, Explicit formulas and recurrence relations for higher order Eulerian polynomials, Indag. Math. 28 (2017), no. 4, 884-891. | ||

In article | |||

[20] | F. Qi and B.-N. Guo, Explicit formulas for derangement numbers and their generating function, J. Nonlinear Funct. Anal. 2016, Article ID 45, 10 pages. | ||

In article | |||

[21] | F. Qi and B.-N. Guo, Some properties of the Hermite polynomials and their squares and generating functions, Preprints 2016, 2016110145, 14 pages. | ||

In article | View Article | ||

[22] | F. Qi and B.-N. Guo, Viewing some ordinary differential equations from the angle of derivative polynomials, Iran. J. Math. Sci. Inform. 14 (2019), no. 2, in press; Preprints 2016, 2016100043, 12 pages. | ||

In article | View Article | ||

[23] | F. Qi, D. Lim, and B.-N. Guo, 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 (2018), in press. | ||

In article | View Article | ||

[24] | F. Qi, D. Lim, and B.-N. Guo, Some identities related to Eulerian polynomials and involving the Stirling numbers, Appl. Anal. Discrete Math. 12 (2018), no. 2, in press. | ||

In article | View Article | ||

[25] | F. Qi, D.-W. Niu, and B.-N. Guo, Simplification of coefficients in differential equations associated with higher order Frobenius-Euler numbers, Preprints 2017, 2017080017, 7 pages. | ||

In article | View Article | ||

[26] | F. Qi, D.-W. Niu, and B.-N. Guo, Simplifying coefficients in differential equations associated with higher or- der Bernoulli numbers of the second kind, Preprints 2017, 2017080026, 6 pages. | ||

In article | View Article | ||

[27] | F. Qi, D.-W. Niu, and B.-N. Guo, 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), in press. | ||

In article | View Article | ||

[28] | F. Qi, X.-L. Qin, and Y.-H. Yao, The generating funtion of the Catalan numbers and lower triangular integer matrices, Preprints 2017, 2017110120, 12 pages. | ||

In article | View Article | ||

[29] | F. Qi, J.-L. Wang, and B.-N. Guo, Notes on a family of inhomogeneous linear ordinary differential equations, Adv. Appl. Math. Sci. 17 (2018), no. 4, 361-368. | ||

In article | |||

[30] | F. Qi, J.-L. Wang, and B.-N. Guo, Simplifying and finding nonlinear ordinary differential equations, ResearchGate Working Paper (2017). | ||

In article | |||

[31] | F. Qi, J.-L. Wang, and B.-N. Guo, Simplifying differential equations concerning degenerate Bernoulli and Euler numbers, Trans. A. Razmadze Math. Inst. 172 (2018), no. 1, 90-94. | ||

In article | |||

[32] | F. Qi and J.-L. Zhao, Some properties of the Bernoulli numbers of the second kind and their generating function, Bull. Korean Math. Soc. 55 (2018). | ||

In article | |||

[33] | F. Qi, Q. Zou, and B.-N. Guo, Some identities and a matrix inverse related to the Chebyshev polynomials of the second kind and the Catalan numbers, Preprints 2017, 2017030209, 25 pages. | ||

In article | View Article | ||

[34] | J. Quaintance and H. W. Gould, 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. | ||

In article | |||

[35] | C.-F. Wei and B.-N. Guo, Complete monotonicity of functions connected with the exponential function and derivatives, Abstr. Appl. Anal. 2014 (2014), Article ID 851213, 5 pages. | ||

In article | |||

[36] | A.-M. Xu and G.-D. Cen, Closed formulas for computing higher-order derivatives of functions involving exponential functions, Appl. Math. Comput. 270 (2015), 136-141. | ||

In article | View Article | ||

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

In article | View Article | ||

[38] | J.-L. Zhao, J.-L. Wang, and F. Qi, Derivative polynomials of a function related to the Apostol-Euler and Frobenius-Euler numbers, J. Nonlinear Sci. Appl. 10 (2017), no. 4, 1345-1349. | ||

In article | |||