Monomial to ultraspherical basis transformation and the zeros of polynomials

TL;DR

This paper proves a conjecture that a linear operator mapping monomials to Legendre polynomials preserves zeros within the interval |x|<1, and extends the result to Jacobi polynomials, enhancing understanding of polynomial zero distributions.

Contribution

It establishes that certain linear operators preserve zeros in the interval for Legendre and Jacobi polynomials, generalizing previous results and confirming a conjecture.

Findings

Operator preserves zeros in |x|<1 for Legendre polynomials

Extension of zero-preservation to Jacobi polynomials

Provides new insights into polynomial zero distributions

Abstract

We examine a result of A. Iserles and E. B. Saff, use it to prove a conjecture of S. Fisk that a linear operator which maps monomials to Legendre polynomials also preserves zeros in the open interval $|x|<1$, and state a more general version of the conjecture for the Jacobi polynomials.