Irreducibility of generalized Hermite–Laguerre polynomials III

For a positive integer n and a real number α, the generalized Laguerre polynomials are defined byLn(α)(x)=∑j=0n(n+α)(n−1+α)⋯(j+1+α)(−x)jj!(n−j)!. These orthogonal polynomials are solutions to Laguerre's Differential Equation   which arises in the treatment of the harmonic oscillator in quantum mechanics. Schur studied these Laguerre polynomials for its interesting algebraic properties. He obtained irreducibility results of Ln(±12)(x) and Ln(±12)(x2) and derived that the Hermite polynomials H2n(x)H2n(x) and H2n+1(x)x are irreducible for each n  . In this article, we extend Schur's result by showing that the family of Laguerre polynomials Ln(q)(x) and Ln(q)(xd) with q∈{±13,±23,±14,±34}, where d is the denominator of q, are irreducible for every n   except when q=14, n=2n=2 where we give the complete factorization. In fact, we derive it from a more general result.