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 Bn,k and their orthogonal companions An,k belong to one single class of Stirling polynomials: An,k=(-1)n-kB-k,-n. 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.