Monomial to ultraspherical basis transformation and the zeros of polynomials

2014/06/26 by Matthew Chasse, Chasse, Matthew
Mathematics · #Algebraic and Geometric Analysis #Holomorphic and Operator Theory #Mathematical functions and polynomials #math.CA #msc:26C10 #msc:30C15

paper · pdf · doi:10.48550/arxiv.1406.6880

8 pages

arxiv created 2014/06/26 · arxiv updated 2014/06/27

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.

Related

• Inequalities for ultraspherical polynomials. Proof of a conjecture of I. Raşa
• Interlacing and monotonicity of zeros of Angelesco-Jacobi polynomials
• A new characterization of ultraspherical, Hermite, and Chebyshev polynomials of the first kind
• Preserving positive polynomials and beyond
• Monotonicity of Zeros of Jacobi-Angelesco polynomials