Irreducibility of extensions of Laguerre polynomials

Abstract

For integers $a_0,a_1,\ldots,a_n$ with $|a_0a_n|=1$ and either $\alpha =u$ with $1\leq u \leq 50$ or $\alpha=u+ \frac{1}{2}$ with $1 \leq u \leq 45$, we prove that $\psi_n^{(\alpha)}(x;a_0,a_1,\cdots,a_n)$ is irreducible except for an explicit finite set of pairs $(u,n)$. Furthermore, all exceptions other than $n=2^{12},\alpha=89/2$ are necessary. The above result with $0\leq\alpha \leq 10$ is due to Filaseta, Finch and Leidy and with $\alpha \in \{-1/2,1/2\}$ due to Schur.