Special 5-term recurrence relations, Banded Toeplitz matrices, and Reality of Zeros

2020/11/26 by Ndikubwayo, Innocent
#12D10 (Primary) 26C10 #30C15 (Secondary) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2011.13258

Abstract

Below we establish the conditions guaranteeing the reality of all the zeros of polynomials P_n(z) in the polynomial sequence \P_n(z)\_n=1^∞ satisfying a five-term recurrence relation P_n(z)= zP_n-1(z) + αP_n-2(z)+βP_n-3(z)+γP_n-4(z), with the standard initial conditions P₀(z) = 1, P_-1(z) = P_-2(z) =P_-3(z) = 0, where α, β, γ are real coefficients, γ≠ 0 and z is a complex variable. We interprete this sequence of polynomials as principal minors of an appropriate banded Teoplitz matrix whose associated Laurent polynomial b(z) is holomorphic in ℂ∖ \0\. We show that when either the critical points of b(z) are all real; or when they are two real and one pair of complex conjugate critical points with some extra conditions on the parameters, the set b^-1(ℝ) contains a Jordan curve with 0 in its interior and in some cases a non-simple curve enclosing 0. The presence of the said curves is necessary and sufficient for every polynomial in the sequence \P_n(z)\_n=1^∞ to be hyperbolic (real-rooted).

Related

• On Hyperbolic Polynomials and Four-term Recurrence with Linear\n Coefficients
• Zeros of polynomials with four-term recurrence and linear coefficients
• Criterion of reality of zeros in a polynomial sequence satisfying a three-term recurrence relation
• Non-real zeros of polynomials in a polynomial sequence satisfying a\n three-term recurrence relation
• Generalizing Tran's Conjecture