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Inverse relations and reciprocity laws involving partial Bell polynomials and related extensions.
TL;DR: In this article, a generalized Lagrange inversion polynomials are introduced that invert functions characterized in a specific way by sequences of constants, and a general reciprocity theorem is established.
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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, Schlomilch-Schlafli 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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Faa di Bruno Hopf algebras
TL;DR: A short review on the Faa di Bruno formulas, and some Hopf algebras associated to them, is given in this article, which allows a short proof of the Lie-Scheffers theorem, and relates the Lagrange inversion formulas with antipodes.
Reciprocal Symmetry via Inverse Series Pairs
TL;DR: Reciprocal series are employed to systematically review convolution sums, orthogonality relations, recurrence relations and reciprocal formulae for several classical number sequences, such as binomial coefficients, Stirling numbers, Bernoulli numbers, and Euler numbers as mentioned in this paper .
Factorial Polynomials and Associated Number Families
Alfred Dr. Schreiber
- 26 Jun 2023
TL;DR: In this paper , two doubly indexed families of polynomials in several indeterminates are considered and they are related to the falling and rising factorials in a similar way as the potential polynomial (introduced by L. Comtet) are related with the ordinary power function.
References
New explicit formulas for the $n$th derivative of composite functions.
TL;DR: In this article, the Taylor coefficients of the parametrically given composite functions are determined by new formulas as explicit functions of the Taylor coefficient of the two component functions, defined implicitly by the parametric representation w = g(t), z = fit.
Some Convolution Identities and an Inverse Relation Involving Partial Bell Polynomials
TL;DR: In this article, an inverse relation and a family of convolution formulas involving partial Bell polynomials are derived from a parametrization of suitable identities that facilitate dealing with nested compositions of partial Bell polynomials.
A characterization of inverse relations
TL;DR: This characterization theorem is used to invert a number of important infinite, lower-triangular matrices, including matrix inversions of Gould and Hsu, Krattenthaler, Carlitz, Bressoud, as well as Andrews' matrix formulation of the Bailey Transform.
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Application of Faà di Bruno's formula in characterization of inverse relations
TL;DR: In this paper, it was shown that a pair of inverse relations can be constructed by the suitable application of Faa di Bruno's formula for finding higher order derivatives of composite functions.
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