Extension of Laguerre polynomials with negative arguments II

Abstract

For $n\ge 3$ and $s\le 92$, it is proved in \cite{ShSi} that, except for finitely many pairs $(n,s),G_1(x)=G_1(x,n,s)$ is either irreducible or linear factor times an irreducible polynomial. If $s\le 30$, we determine here explicitely the set of pairs $(n,s)$ in the above assertion. This implies a new proof of the result of Nair and Shorey \cite{NaSh1} that $G_1(x)$ is irreducible for $s\le 22$.