Submitted:
24 August 2017
Posted:
25 August 2017
You are already at the latest version
Abstract
In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.
Keywords:
Bell polynomial
; Bell number
; Bell polynomial of the second kind
; higher order derivative
; generating function
; Faa di Bruno formula
; inversion theorem
; Stirling number of the first kind
; Stirling number of the second kind
; explicit formula
; inversion formula
; logarithmically absolute monotonicity
; logarithmically complete monotonicity
; determinantal inequality
; product inequality
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
