Irreducibility of extended Laguerre polynomials

Let m >= 1 and am be integers. Let alpha be a rational number which is not a negative integer such that alpha = 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 ai(x) with 0 <= i <= m - 1 belonging to Z[x] be polynomials having degree less than deg phi(x). Assume that the content of (ama0(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) = 1 / m!(am phi(x)(m) +& sum; (m-1)(j=0)b( j)a(j)(x)phi(x)j) are irreducible over the rationals for each alpha is an element of{0, 1, 2, 3, 4} unless (m,alpha) is an element of{(1, 0), (2, 2), (4, 4), (6, 4)}, where b(j) is given by b(j )= m (j) (m + alpha)(m - 1 + alpha)& ctdot;(j + 1 + alpha) for 0 <= j <= m - 1. Further, we show that L-m,alpha(phi)(x) is irreducible over rationals for all but finitely many m. For proving our results, we use a beautiful result of Theorem 1.3 of Jindal and Khanduja [An extension of Schur's irreducibility result, J. Algebra 664 (2025) 398-409] which was the main tool in extending Schur's irreducibility result. We illustrate our results through examples.