Convexity of the zeros of some orthogonal polynomials and related functions

TL;DR

This paper investigates the convexity properties of zeros of Laguerre, Jacobi, and ultraspherical polynomials, providing bounds and using transformations that preserve zeros to deepen understanding of their distribution.

Contribution

It establishes convexity intervals and bounds for zeros of classical orthogonal polynomials and related functions, extending Sturm's convexity theorem applications.

Findings

Convexity intervals for zeros of Laguerre, Jacobi, and ultraspherical polynomials identified.

Upper and lower bounds for distances between consecutive zeros derived.

Transformations preserving zeros used to analyze convexity properties.

Abstract

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases.