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Stirling Polynomials in Several Indeterminates

Alfred Schreiber

ISBN 978-3-8325-5250-3
160 pages, year of publication: 2021
price: 29.00 €
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.

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.

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Keywords:
  • Algebra
  • Combinatorics
  • Polynomials
  • Inversion
  • Reciprocity

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