Connections between discriminants and the root distribution of polynomials with rational generating function

Abstract

Let $H_m(z)$ be a sequence of polynomials whose generating function ${\sum }_{m=0}^{\infty }{H}_m(z)t^m$ is the reciprocal of a bivariate polynomial $D(t,z)$. We show that in the three cases $D(t,z)=1+B(z)t+A(z)t^2$, $D(t,z)=1+B(z)t+A(z)t^3$ and $D(t,z)=1+B(z)t+A(z)t^4$, where $A(z)$ and $B(z)$ are any polynomials in $z$ with complex coefficients, the roots of $H_m(z)$ lie on a portion of a real algebraic curve whose equation is explicitly given. The proofs involve the $q$-analogue of the discriminant, a concept introduced by Mourad Ismail.

References in corpus (1)

• Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

Cited by in corpus (11)

• The root distribution of polynomials with a three-term recurrence
• Zeros of polynomials with four-term recurrence
• Polynomials with rational generating functions and real zeros
• Around a Conjecture of K. Tran
• Exact real time dynamics with free fermions in disguise
• Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices
• Generalizing Tran's Conjecture
• On Hyperbolic Polynomials and Four-term Recurrence with Linear Coefficients
• Piecewise interlacing zeros of polynomials
• On the zeros of polynomials generated by rational functions with a hyperbolic polynomial type denominator
• Root geometry of polynomial sequences III: Type $(1,1)$ with positive coefficients