Irreducibility of Generalised Laguerre Polynomials for Large Integer Parameters

Source abstract

For a positive integer $n$, we study the irreducibility of the generalised Laguerre polynomials $$L_n^{(a)}(x) = \frac{1}{n!}\sum_{j=0}^{n} \binom{n}{j} \frac{(n+a)!}{(j+a)!}(-1)^j x^j,$$ where $a \in \mathbb{Z}_{\ge 0}$. Earlier works established that these polynomials are irreducible over $\mathbb{Q}$ for $0 \le a \le 50$. In this paper, we substantially strengthen these results by establishing that $L_n^{(a)}(x)$ are irreducible for all $a \le 1000$. We derive sharper constraints on potential integer roots and make systematic use of the geometry of the associated Newton polygons. This approach allows us to extend the irreducibility range significantly while keeping explicit computations to a minimum.