Extensions of Schur's irreducibility results

TL;DR

This paper proves that generalized Laguerre polynomials are irreducible for most parameter values, with only finitely many exceptions, extending Schur's classical irreducibility results.

Contribution

It generalizes Schur's irreducibility results to a broader class of Laguerre polynomials with parameters up to 50, identifying all exceptions.

Findings

Most generalized Laguerre polynomials with 0 ≤ α ≤ 50 are irreducible.

Finitely many pairs (n, α) are exceptions to irreducibility.

Exceptions are proven to be necessary and are explicitly characterized.

Abstract

We prove that the generalised Laguerre polynomials $L_{n}^{(\alpha)}(x)$ with $0\le \al\le 50$ are irreducible except for finitely many pairs $(n, \al)$ and that these exceptions are necessary. In fact it follows from a more general statement.