On the irreducibility of extended Laguerre Polynomials

Let $m\geq 1$ 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 $\Z[x]$ be a monic polynomial which is irreducible modulo all the primes less than or equal to $vm+u$. Let $a_i(x)$ with $0\leq i\leq m-1$ belonging to $\Z[x]$ be polynomials having degree less than $\deg\phi(x)$. Assume that the content of $(a_ma_0(x))$ is not divisible by any prime less than or equal to $vm+u$. In this paper, we prove that the polynomials $L_{m,\alpha}^{\phi}(x) = \frac{1}{m!}(a_m\phi(x)^m+\sum\limits_{j=0}^{m-1}b_ja_j(x)\phi(x)^j)$ are irreducible over the rationals for all but finitely many $m$, where $b_j = \binom{m}{j}(m+\alpha)(m-1+\alpha)\cdots (j+1+\alpha)~~~\mbox{ for }0\leq j\leq m-1$. Further, we show that $L_{m,\alpha}^{\phi}(x)$ is irreducible over rationals for each $\alpha \in \{0, 1, 2, 3, 4\}$ unless $(m, \alpha) \in \{ (1,0), (2,2), (4,4),(6,4)\}.$ 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.