Rational convolution roots of isobaric polynomials

Abstract

In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under the convolution product, in terms of determinants and permanents, respectively. The underlying root formulas for GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier paper by MacHenry and Tudose, make use of two types of operators. These operators are derived from the generating functions for Stirling numbers of the first and second kind. Hence, we call them Stirling operators. In order to construct matrix representations of the roots of GFPs, we use Stirling operators of the first kind. We give explicit examples to show how Stirling operators of the second kind appear in low degree cases for the WIP-roots. As a consequence of the matrix construction we have matrix representations of multiplicative arithmetic functions under the Dirichlet product into its divisible closure.