Inverse relations and reciprocity laws involving partial Bell polynomials and related extensions
Abstract
The objective of this paper is, in the main, twofold: Firstly, to develop an algebraic setting for dealing with Bell polynomials and related extensions. Secondly, based on the author's previous work on multivariate Stirling polynomials (2015), to present a number of new results related to different types of inverse relationships, among these (1) the use of multivariable Lah polynomials for characterizing self-orthogonal families of polynomials that can be represented by Bell polynomials, (2) the introduction of `generalized Lagrange inversion polynomials' that invert functions characterized in a specific way by sequences of constants, (3) a general reciprocity theorem according to which, in particular, the partial Bell polynomials and their orthogonal companions belong to one single class of Stirling polynomials: . Moreover, of some numerical statements (such as Stirling inversion, Schl\"omilch-Schl\"afli formulas) generalized polynomial versions are established. A number of well-known theorems (Jabotinsky, Mullin-Rota, Melzak, Comtet) are given new proofs.
Keywords
Cite
@article{arxiv.2009.09201,
title = {Inverse relations and reciprocity laws involving partial Bell polynomials and related extensions},
author = {Alfred Schreiber},
journal= {arXiv preprint arXiv:2009.09201},
year = {2021}
}
Comments
73 pages. The article continues the research reported by the author in his paper "Multivariate Stirling polynomials of the first and second kind", Discrete Mathematics 338 (2015), 2462-2484. Preprint version: arXiv:1311.5067