Extension of Laguerre polynomials with negative arguments

We consider the irreducibility of polynomial $L_n^{(α)} (x)$ where $α$ is a negative integer. We observe that the constant term of $L_n^{(α)} (x)$ vanishes if and only if $n \geq |α| = -α$. Therefore we assume that $α= -n-s-1$ 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.

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