Exploring the zeros of real self-reciprocal polynomials by Chebyshev polynomials

TL;DR

This paper investigates the zeros of specific real self-reciprocal polynomials linked to Chebyshev quasi-orthogonal polynomials, analyzing their distribution, simplicity, and monotonicity around the unit circle and real line.

Contribution

It introduces classes of such polynomials with at most two zeros outside the unit circle and explores their zero distribution properties.

Findings

Zeros are mostly on the unit circle or real line.

Zeros outside the unit circle are limited to at most two.

Zeros exhibit monotonicity and simplicity properties.

Abstract

In this paper we present some classes of real self-reciprocal polynomials with at most two zeros outside the unit circle which are connected with a Chebyshev quasi-orthogonal polynomials of order one. We investigated the distribution, simplicity and monotonicity of their zeros around the unit circle and real line.