On the irreducibility of extended Laguerre Polynomials

Let $m\ge 1$ and $a_m$ be integers. Let $\alpha$ be a rational number which is not a negative integer such that $\alpha =\frac{u}{v}$ with $\gcd (u,v)=1,v>0$. Let $\phi (x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all the primes less than or equal to $vm+u$. Let $a_i(x)$ with $0\le i\le m-1$ belonging to $\mathbb{Z}[x]$ be polynomials having degree less than $\mathrm{deg}\phi (x)$. Assume that the content of $(a_m{a}_0(x))$ is not divisible by any prime less than or equal to $vm+u$. In this paper, we prove that the polynomials $L_{m,\alpha }^{\phi }(x)=\frac{1}{m!}(a_m\phi (x{)}^m+{\sum }_{j=0}^{m-1}{b}_j{a}_j(x)\phi (x{)}^j)$ are irreducible over the rationals for all but finitely many $m$, where $b_j=\left(\genfrac{}{}{0px}{}{m}{j}\right)(m+\alpha )(m-1+\alpha )\cdots (j+1+\alpha )for0\le j\le m-1$. Further, we show that $L_{m,\alpha }^{\phi }(x)$ is irreducible over rationals for each $\alpha \in \{0,1,2,3,4\}$ unless $(m,\alpha )\in \{(1,0),(2,2),(4,4),(6,4)\}\mathrm{.}$ For proving our results, we use the notion of $\phi$-Newton polygon and some results from analytic number theory. We illustrate our results through examples.