Zeros of a binomial combination of Chebyshev polynomials

TL;DR

This paper investigates the zeros of polynomials generated by a binomial combination of Chebyshev polynomials, showing that only finitely many zeros lie outside the interval (-1,1), regardless of polynomial degree.

Contribution

It establishes a uniform bound on the number of zeros outside (-1,1) for polynomials generated from a binomial combination of Chebyshev polynomials.

Findings

Number of zeros outside (-1,1) is bounded independently of degree m.

Zeros are mostly confined within the interval (-1,1).

Provides insight into the zero distribution of these polynomial sequences.

Abstract

For $0<\alpha<1$, we study the zeros of the sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ generated by the reciprocal of $(1-t)^{\alpha}(1-2zt+t^{2})$, expanded as a power series in $t$. Equivalently, this sequence is obtained from a linear combination of Chebyshev polynomials whose coefficients have a binomial form. We show that the number of zeros of $P_{m}(z)$ outside the interval $(-1,1)$ is bounded by a constant independent of $m$.