Stirling Polynomials in Several Indeterminates
Alfred Schreiber
Diese Publikation zitieren
Alfred Schreiber, Stirling Polynomials in Several Indeterminates (2021), Logos Verlag, Berlin, ISBN: 9783832586058
Beschreibung / Abstract
The classical exponential polynomials, today commonly named after
E. ,T. Bell, have a wide range of remarkable applications in
Combinatorics, Algebra, Analysis, and Mathematical Physics. Within the
algebraic framework presented in this book they appear as structural
coefficients in finite expansions of certain higher-order derivative
operators. In this way, a correspondence between polynomials and
functions is established, which leads (via compositional inversion) to
the specification and the effective computation of orthogonal
companions of the Bell polynomials. Together with the latter, one
obtains the larger class of multivariate `Stirling polynomials'. Their
fundamental recurrences and inverse relations are examined in detail
and shown to be directly related to corresponding identities for the
Stirling numbers. The following topics are also covered: polynomial
families that can be represented by Bell polynomials; inversion
formulas, in particular of Schlömilch-Schläfli type; applications to
binomial sequences; new aspects of the Lagrange inversion, and, as a
highlight, reciprocity laws, which unite a polynomial family and that
of orthogonal companions. Besides a
Mathematica(R) package and an extensive
bibliography, additional material is compiled in a number of notes and
supplements.
E. ,T. Bell, have a wide range of remarkable applications in
Combinatorics, Algebra, Analysis, and Mathematical Physics. Within the
algebraic framework presented in this book they appear as structural
coefficients in finite expansions of certain higher-order derivative
operators. In this way, a correspondence between polynomials and
functions is established, which leads (via compositional inversion) to
the specification and the effective computation of orthogonal
companions of the Bell polynomials. Together with the latter, one
obtains the larger class of multivariate `Stirling polynomials'. Their
fundamental recurrences and inverse relations are examined in detail
and shown to be directly related to corresponding identities for the
Stirling numbers. The following topics are also covered: polynomial
families that can be represented by Bell polynomials; inversion
formulas, in particular of Schlömilch-Schläfli type; applications to
binomial sequences; new aspects of the Lagrange inversion, and, as a
highlight, reciprocity laws, which unite a polynomial family and that
of orthogonal companions. Besides a
Mathematica(R) package and an extensive
bibliography, additional material is compiled in a number of notes and
supplements.
Beschreibung
Alfred Schreiber was born in Düsseldorf (Germany) in 1946, studied mathematics and philosophy, and received his doctorate from the University of Cologne in 1974. He held post-doc positions at the Pädagogische Hochschule Rheinland and, after habilitation in 1981, as ``Privatdozent'' at the RWTH Aachen. After a two-year interlude as a software developer in the private sector, he received a full professorship at the University of Flensburg in 1986, where he taught and researched for more than two decades. His interests include epistemological topics in mathematics education, media informatics, enumerative combinatorics, and a wide range of interdisciplinary aspects of mathematics. He is also active as a literary author and translator. Two anthologies of mathematical poems edited by him have been published in 2010/12 by Springer Verlag.
Inhaltsverzeichnis
- BEGINN
- I Multivariate Stirling Polynomials
- 1 Introduction
- 2 Function algebra with derivation
- 3 Expansion of higher-order derivatives
- 4 A brief summary on Bell polynomials
- 5 Inversion formulas and recurrences
- 6 Explicit formulas for Sn;k
- 7 Remarks on Lagrange inversion
- 8 Concluding remarks
- II Inverse Relations and Reciprocity Laws
- 1 Introduction
- 2 Basic notions and preliminaries
- 3 Polynomials from Taylor coeXcients
- 4 Composition rules
- 5 Representation by Bell polynomials
- 6 Applications to binomial sequences
- 7 Lagrange inversion polynomials
- 8 Reciprocity theorems
- Appendix A Mathematica Package
- Bibliography