负变元Laguerre多项式的推广
Extension of Laguerre polynomials with negative arguments
摘要：
本文考虑多项式$L_n^{(α)} (x) $的不可约性，其中$α$为负整数.我们观察到$L_n^{(α)} (x) $的常数项为零当且仅当$n \geq |α| = -α$。因此，我们假设$α= -n-s-1$，其中$s$是非负整数.设$$ g(x) = (-1)^n L_n^{(-n-s-1)}(x) = \sum\limits_{j=0}^{n} a_j \frac{x^j}{j!} $$和更一般的多项式，设$$ G(x) = \sum\limits_{j=0}^{n} a_j b_j \frac{x^j}{j!} $$其中$b_j$和$0 \leq j \leq n$是整数，使得$|b_0| = |b_n| = 1$。Schur是第一个证明$g(x)$对$s=0$不可约化的人。证明了$g(x)$对$0 \leq s \leq 60$是不可约的。本文用另一种方法证明了：除了许多明确给出的可能性外，$G(x)$是不可约的或$G(x)$是线性因子乘不可约多项式。这是当$G(x)$具有度因子$k \geq 2$和$(n,k,s) \neq (10,5,4)$时估计$s > 1.9 k$的结果。这使Shorey和Tijdeman以及Nair和Shorey的早期估计更加尖锐。
英文摘要：
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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