Spectral density functions of bivariable stable polynomials

TL;DR

This paper investigates the spectral density functions of bivariate stable polynomials, focusing on their Fourier coefficient asymptotics and involving hypergeometric functions to derive new mathematical results.

Contribution

It provides new insights into the spectral density of bivariate stable polynomials and introduces hypergeometric functions into the analysis of Fourier coefficient asymptotics.

Findings

Derived new asymptotic formulas for Fourier coefficients

Connected spectral density functions with hypergeometric functions

Enhanced understanding of multivariable polynomial stability

Abstract

The relationship between a stable multivariable polynomial $p(z)$ and the Fourier coefficients of its spectral density function $1/|p(z)|^2$, is further investigated. In this paper we focus on the radial asymptotics of the Fourier coefficients for a specific choice of a two variable polynomial. Hypergeometric functions appear in the analysis, and new results are derived for these as well.