Preprint Article Version 1 This version is not peer-reviewed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)

How to cite: Qi, F.; Niu, D.; Lim, D.; Guo, B. Generalizations of the Bell Numbers and Polynomials and Their Properties. Preprints 2017, 2017080090 (doi: 10.20944/preprints201708.0090.v1). Qi, F.; Niu, D.; Lim, D.; Guo, B. Generalizations of the Bell Numbers and Polynomials and Their Properties. Preprints 2017, 2017080090 (doi: 10.20944/preprints201708.0090.v1).

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

Readers' Comments and Ratings (1)

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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