Irreducibility of extensions of Laguerre Polynomials

Abstract

For integers $a_0,a_1,\dots ,a_n$ with $|a_0{a}_n|=1$ and either $\alpha =u$ with $1\le u\le 50$ or $\alpha =u+\frac{1}{2}$ with $1\le u\le 45$, 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 the exceptions other than $n=2^{12},\alpha =89/2$ are necessary. The above result with $0\le \alpha \le 10$ is due to Filaseta, Finch and Leidy and with $\alpha \in \{-1/2,1/2\}$ due to Schur.