On the zeros of polynomials generated by rational functions with a hyperbolic polynomial type denominator

2016/06/22 by Tamás Forgács, Forgács, Tamás, Khang Tran +1
Mathematics · #11C08 #26C10 #30C15 #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1606.07125

openalex publication_date 2016/06/22 · openalex created_date 2016/07/22 · openalex updated_date 2026/07/28

Abstract

This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a denominator of the form G(z,t)=P(t)+zt^r, where the zeros of P are positive and real. We show that every member of a family of such generating functions - parametrized by the degree of P and r - gives rise to a sequence of polynomials \H_m(z)\_m=0^∞ that is eventually hyperbolic. Moreover, when P(0)>0 the real zeros of the polynomials H_m(z) form a dense subset of an interval I⊂ℝ⁺, whose length depends on the particular values of the parameters in the generating function.

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