Extension of irreducibility results on generalized Laguerre polynomials $L_n^{(-n-s-1)}(x)$

Let $n$ be a positive integer. We consider the irreducibility of generalized Laguerre polynomials of the form $$L_n^{(-n-s-1)}(x)=\displaystyle\sum_{j=0}^{n}(-1)^n\binom{n+s-j}{n-j}\frac{x^j}{j!}.$$ For different values of $s$, this family gives polynomials which are of great interest. It was proved earlier that for $0 \leq s \leq 60$, these polynomials are irreducible over $\mathbb{Q}$, and their Galois groups are shown to be $A_n$ or $S_n$. In this paper, we prove that $L_n^{(-n-s-1)}(x)$ is irreducible for each $s \leq 92$. Also, we prove that $L_n^{(-n-s-1)}(x)$ has no linear factor for each $93 \leq s \leq 100$. Furthermore, assuming the irreducibility of $L_n^{(-n-s-1)}(x)$ for $93 \leq s \leq 100$, we determine that the Galois group of $L_n^{(-n-s-1)}(x)$ is either $A_n$ or $S_n$ for each $61\leq s \leq 100$.