Exceptional Laguerre polynomials

TL;DR

This paper systematically constructs exceptional Laguerre polynomials using partitions, explores their properties, and provides new asymptotic results on the distribution of their zeros as the degree increases.

Contribution

It introduces a systematic construction method for exceptional Laguerre polynomials using partitions and derives new asymptotic results on their zeros.

Findings

Partition-based expression of exceptional Laguerre polynomials

Asymptotic behavior of zeros as degree tends to infinity

Restatement of known properties in partition terms

Abstract

The aim of this paper is to present the construction of exceptional Laguerre polynomials in a systematic way, and to provide new asymptotic results on the location of the zeros. To describe the exceptional Laguerre polynomials we associate them with two partitions. We find that the use of partitions is an elegant way to express these polynomials and we restate some of their known properties in terms of partitions. We discuss the asymptotic behavior of the regular zeros and the exceptional zeros of exceptional Laguerre polynomials as the degree tends to infinity.