Reflections on symmetric polynomials and arithmetic functions
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.
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