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 $B_{n,k}$ and their orthogonal companions $A_{n,k}$ belong to one single class of Stirling polynomials: $A_{n,k}=(-1{)}^{n-k}{B}_{-k,-n}$. Moreover, of some numerical statements (such as Stirling inversion, Schlömilch-Schläfli formulas) generalized polynomial versions are established. A number of well-known theorems (Jabotinsky, Mullin-Rota, Melzak, Comtet) are given new proofs.

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