Rational Convolution Roots of Isobaric Polynomials

TL;DR

This paper develops matrix representations for the rational roots of generalized Fibonacci polynomials and weighted isobaric polynomials using Stirling operators, linking them to determinants, permanents, and arithmetic functions.

Contribution

It introduces new matrix representations of GFP and WIP roots using Stirling operators, connecting polynomial roots with matrix theory and arithmetic functions.

Findings

Matrix representations of GFP and WIP roots via determinants and permanents.

Explicit examples demonstrating Stirling operators in low degree cases.

Representation of multiplicative arithmetic functions under Dirichlet product.

Abstract

In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under 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 kind and second kind. Hence we call them Stirling operators. To construct matrix representations of the roots of GFPs, we use the Stirling operators of the first kind. We give explicit examples to show how the Stirling operators of the second kind appear in the 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.