Polynomials with rational generating functions and real zeros

Abstract

This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a binomial-type denominator. We show that every member of a two-parameter family consisting of such generating functions gives rise to a sequence of polynomials $\{P_m(z){\}}_{m=0}^{\infty }$ that is eventually hyperbolic. Moreover, the real zeros of the polynomials $P_m(z)$ form a dense subset of an interval $I\subset {\mathbb{R}}^{+}$, whose length depends on the particular values of the parameters in the generating function.

References in corpus (3)

• Connections between discriminants and the root distribution of polynomials with rational generating function
• The root distribution of polynomials with a three-term recurrence
• Hermite multiplier sequences and their associated operators

Cited by in corpus (5)

• Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials
• Zeros of polynomials with four-term recurrence
• On the zeros of polynomials generated by rational functions with a hyperbolic polynomial type denominator
• Totally nonnegative matrices, chain enumeration and zeros of polynomials
• Non-self-adjoint Toeplitz matrices whose principal submatrices have real spectrum