  We consider the irreducibility of polynomial $L_n^{(\alpha)} (x) $ where
$\alpha$ is a negative integer. We observe that the constant term of
$L_n^{(\alpha)} (x) $ vanishes if and only if $n \geq |\alpha| = -\alpha$.
Therefore we assume that $\alpha = -n-s-1$ where $s$ is a non-negative integer.
Let $$ g(x) = (-1)^n L_n^{(-n-s-1)}(x) = \sum\limits_{j=0}^{n} a_j
\frac{x^j}{j!} $$ and more general polynomial, let $$ G(x) =
\sum\limits_{j=0}^{n} a_j b_j \frac{x^j}{j!} $$ where $b_j$ with $0 \leq j \leq
n$ are integers such that $|b_0| = |b_n| = 1$. Schur was the first to prove the
irreducibility of $g(x)$ for $s=0$. It has been proved that $g(x)$ is
irreducibile for $0 \leq s \leq 60$. In this paper, by a different method, we
prove : Apart from finitely many explicitely given posibilities, either $G(x)$
is irreducible or $G(x)$ is linear factor times irreducible polynomial. This is
a consequence of the estimate $s > 1.9 k$ whenever $G(x)$ has a factor of
degree $k \geq 2$ and $(n,k,s) \neq (10,5,4)$. This sharpens earlier estimates
of Shorey and Tijdeman and Nair and Shorey.
