包含部分Bell多项式的逆关系和互反律及相关推广
Inverse relations and reciprocity laws involving partial Bell polynomials and related extensions
摘要：
本文的目的是，在主要的，双重的：首先，开发一个代数设置处理贝尔多项式和相关的扩展。其次，基于作者以前关于多元Stirling多项式的工作（2015），提出了一些与不同类型的逆关系相关的新结果，其中（1）使用多元Lah多项式来表征可以由Bell多项式表示的自正交多项式族，（2）引入了“广义拉格朗日反演多项式”，该多项式反演了以特定方式由常数序列表征的函数，（3）一般互易定理，根据该定理，特别是，部分贝尔多项式$B_{n,k}$和它们的正交同伴$A_{n,k}$属于单一类的斯特林多项式：$A_{n,k}=(-1)^{n-k}B_{-k,-n}$。此外，还建立了一些数值表述（如Stirling反演、Schlömilch-Schläfli公式）的广义多项式形式。一些著名的定理（Jabotinsky，Mullin-Rota，Melzak，Comtet）给出了新的证明。
英文摘要：
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.
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