Preprint Article Version 1 This version is not peer-reviewed

Generalizations of the Bell Numbers and Polynomials and Their Properties

Version 1 : Received: 25 August 2017 / Approved: 26 August 2017 / Online: 26 August 2017 (09:12:05 CEST)

A peer-reviewed article of this Preprint also exists.

Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, Some properties and an application of multivariate exponential polynomials, Mathematical Methods in the Applied Sciences (2020), no.~6, 2967-2983; available online at https://doi.org/10.1002/mma.6095. Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, Some properties and an application of multivariate exponential polynomials, Mathematical Methods in the Applied Sciences (2020), no.~6, 2967-2983; available online at https://doi.org/10.1002/mma.6095.

Journal reference: Mathematical Methods in the Applied Sciences 2020, 43, 2967-2983
DOI: 10.1002/mma.6095

Abstract

In the paper, the authors present unified generalizations for the Bell numbers and polynomials, establish explicit formulas and inversion formulas for these generalizations in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem connected with the Stirling numbers of the first and second kinds, construct determinantal and product inequalities for these generalizations with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations.

Subject Areas

Bell number; Bell polynomial; generalization; explicit formula; inversion formula; inversion theorem; Stirling number; Bell polynomial of the second kind; determinantal inequality; product inequality; completely monotonic function; logarithmic convexity

Comments (3)

Comment 1
Received: 31 August 2017
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the first and corresponding author of this preprint.
Comment: Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, ''A unified generalization of the Bell numbers and the Touchard polynomials and its properties'', ResearchGate Working Paper (2017), available online at https://doi.org/10.13140/RG.2.2.36733.05603
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Comment 2
Received: 11 June 2018
Commenter: Fng Qi
Commenter's Conflict of Interests: I am the first and corresponding author of this preprint.
Comment: This preprint is the original version of the following preprint:

Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, Some properties and an application of multivariate exponential polynomials, HAL archives (2018), available online at https://hal.archives-ouvertes.fr/hal-01745173
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Comment 3
Received: 6 March 2020
Commenter: Feng Qi (Click to see Publons profile: )
Commenter's Conflict of Interests: I am the first and corresponding author
Comment: This preprint has been formally published as

Feng Qi, Da-Wei Niu, Dongkyu Lim, and Bai-Ni Guo, Some properties and an application of multivariate exponential polynomials, Mathematical Methods in the Applied Sciences 43 (2020), no. 6, 2967--2983; available online at doi.org/10.1002/mma.6095https://doi.org/10.1002/mma.6095.
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