The Laguerre polynomials preserve real-rootedness

TL;DR

This paper proves that the linear transformation mapping monomials to Laguerre polynomials maintains the property of having all roots real, highlighting a significant stability feature of Laguerre polynomials.

Contribution

It establishes that Laguerre polynomial transformations preserve real-rootedness, a property not previously confirmed for this class of polynomials.

Findings

Laguerre polynomial transformation preserves real-rootedness

The result extends understanding of polynomial root stability

Implications for analysis and combinatorics

Abstract

The linear transformation that sends $x^n$ to the n'th Laguerre polynomial preserves real-rootedness.