Irreducibility of extended Laguerre polynomials

Let [Formula: see text] and [Formula: see text] be integers. Let [Formula: see text] be a rational number which is not a negative integer such that [Formula: see text] with [Formula: see text]. Let [Formula: see text] belonging to [Formula: see text] be a monic polynomial which is irreducible modulo all the primes less than or equal to [Formula: see text]. Let [Formula: see text] with [Formula: see text] belonging to [Formula: see text] be polynomials having degree less than [Formula: see text]. Assume that the content of [Formula: see text] is not divisible by any prime less than or equal to [Formula: see text]. In this paper, we prove that the polynomials [Formula: see text] are irreducible over the rationals for each [Formula: see text] unless [Formula: see text], where [Formula: see text] is given by [Formula: see text]. Further, we show that [Formula: see text] is irreducible over rationals for all but finitely many [Formula: see text]. 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.