Preprint Article Version 1 This version is not peer-reviewed

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

Version 1 : Received: 4 August 2017 / Approved: 4 August 2017 / Online: 4 August 2017 (15:47:44 CEST)

How to cite: Qi, F.; Niu, D.; Guo, B. Simplification of Coefficients in Differential Equations Associated with Higher Order Frobenius–Euler Numbers. Preprints 2017, 2017080017 (doi: 10.20944/preprints201708.0017.v1). Qi, F.; Niu, D.; Guo, B. Simplification of Coefficients in Differential Equations Associated with Higher Order Frobenius–Euler Numbers. Preprints 2017, 2017080017 (doi: 10.20944/preprints201708.0017.v1).

Abstract

In the paper, by virtue of the Fa`a di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius–Euler numbers.

Subject Areas

simplification; coefficient; ordinary differential equation; higher order Frobenius–Euler number; Fa`a di Bruno formula; Bell polynomial of the second kind; inversion formula

Comments (2)

Comment 1
Received: 21 March 2019
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the first and corresponding author
Comment: This preprint has been formally published as follows:

Feng Qi, Da-Wei Niu, and Bai-Ni Guo, Simplification of coefficients in differential equations associated with higher order FrobeniusEuler numbers, Tatra Mountains Mathematical Publications 72 (2018), 6776; Available online at https://doi.org/10.2478/tmmp-2018-0022
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Comment 2
Received: 17 September 2019
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the first and corresponding author
Comment: A revised version of this preprint has been formally published as

Feng Qi, Da-Wei Niu, and Bai-Ni Guo, Simplification of coefficients in differential equations associated with higher order Frobenius—Euler numbers, Tatra Mountains Mathematical Publications 72 (2018), 67—76; available online at https://doi.org/10.2478/tmmp-2018-0022
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