The asymptotic zero distribution of Lommel polynomials as polynomials of the order with a variable complex argument

TL;DR

This paper analyzes the asymptotic distribution of roots of Lommel polynomials with a variable complex argument, revealing their accumulation on specific curves and providing formulas for their density and support.

Contribution

It establishes the weak limit of root-counting measures for Lommel polynomials and derives explicit formulas for the supporting curves and density, advancing understanding of their asymptotic behavior.

Findings

Roots accumulate on specific curves in the complex plane.

Existence of a weak limit for root-counting measures is proven.

Formulas for the support and density of roots are derived.

Abstract

We study the asymptotic distribution of roots of Lommel polynomials as polynomials of the order with a variable and purely imaginary argument. The roots are complex and accumulate on certain curves in the complex plane. We prove existence of the weak limit of corresponding root-counting measures and deduce formulas for the supporting curves and density. The obtained result represents a solvable example of a more general problem which is still open. Numerical illustrations of the main result are also involved.