论文标题
涉及部分铃铛多项式和相关扩展的反关系和互惠法
Inverse relations and reciprocity laws involving partial Bell polynomials and related extensions
论文作者
论文摘要
本文的目的是主要是双重的：首先，开发一个代数设置，用于处理钟形多项式和相关扩展。其次，基于作者先前关于多元stirling多项式的工作（2015），介绍了许多与不同类型的逆关系有关的新结果，其中（1）使用多种多样的多项式来表征多项式的自我定位族的自我定位家庭，这些多项式可以用Bell interionals（bell oftertion）代表多个bell polynomials（2）``2）''（2）``2）''（2）``2）``2）。 （3）以特定方式以特定的方式进行特征，（3）尤其是根据该定理的一般互惠定理，尤其是部分钟形多项式$ b_ {n，k} $及其正交伴侣$ a_ {n，k} $属于一个单一类的Stirling polynomials： $ a_ {n，k} =（ - 1）^{n-k} b _ { - k，-n} $。此外，建立了一些数值语句（例如Stirling倒置，Schlömilch-Schläfli公式）广义多项式版本。许多知名定理（Jabotinsky，Mullin-Rota，Melzak，Comtet）得到了新的证明。
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.
