Reflections on symmetric polynomials and arithmetic functions
type: Item
instance of: scholarly article
title: Reflections on symmetric polynomials and arithmetic functions
zbMATH Open document ID: 1079.05097
author: Trueman MacHenry
author: Geanina Tudose
published in: Rocky Mountain Journal of Mathematics
publication date: 2005-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0106213
review text: An isobaric polynomial is a polynomial \(\sum_\alpha A(\alpha) t_1^{\alpha_1} \cdots t_n^{\alpha_n}\) where \(\sum j \alpha_j\) is the same for all \(\alpha\). There is a canonical isomorphism between the isobaric polynomials and the symmetric polynomials. The authors study families of isobaric polynomials indexed by rational weight vectors; examples include the images of the complete symmetric and power sum symmetric polynomials, which are the generalized Fibonacci and Lucas polynomials studied in [\textit{T. MacHenry}, Fibonacci Q. 38, 167--173 (2000; Zbl 0963.11010)]. A product on weighted isobaric families is defined by convolution, and the \(q\)th convolution root of a family is computed. This leads to a generalization of the results of Carroll and Gioia on the \(q\)th roots of the core functions, which are a subgroup of the multiplicative arithmetic functions.
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 11N99
zbMATH DE Number: 2207775
zbMATH Keywords: symmetric functions
zbMATH Keywords: isobaric polynomials
zbMATH Keywords: multiplicative arithmetic functions
reviewed by: David Grabiner
MaRDI profile type: Publication
OpenAlex ID: W1994181109
arXiv ID: math/0106213
cites work: Q3778055
cites work: Q3129746
cites work: Q4002278
cites work: Q4328336
cites work: A subgroup of the group of units in the ring of arithmetic functions
cites work: Q4943122
cites work: Q4488957
cites work: Differential operators and weighted isobaric polynomials
cites work: EXTENDING GAUSSIAN HYPERGEOMETRIC SERIES TO THE p-ADIC SETTING
cites work: Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen
cites work: Q5184913
cites work: The Theory of Multiplicative Arithmetic Functions
DOI: 10.1216/RMJM/1181069713
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description: scientific article