Irreducibility of generalized Laguerre polynomials Ln(12+u)(x) with integer u

Generalized Laguerre polynomials Ln(α)(x) are classical orthogonal polynomial sequences that play an important role in various branches of analysis and mathematical physics. Schur (1929) was the first to study the algebraic properties of these polynomials by proving that Ln(α)(x) where α∈{0,1,−n−1}α∈{0,1,−n−1} are irreducible. For α=u+12 with integer u   satisfying 1≤u≤451≤u≤45, we prove that Ln(α)(x) and Ln(α)(x2) of degrees n and 2n  , respectively, are irreducible except when (u,n)=(10,3)(u,n)=(10,3) where we give a factorization. The cases u=−1,0u=−1,0 are due to Schur. Further we consider more general polynomials Gα(x)Gα(x) and Gα(x2)Gα(x2) of degrees n and 2n  , respectively, and prove that they are either irreducible or have a factor of degree in {1,n−1}{1,n−1}, {1,2,2n−2,2n−1}{1,2,2n−2,2n−1}, respectively, except for an explicitly given finite set of pairs (u,n)(u,n). We also show that these exceptional pairs other than one for Gα(x)Gα(x) and six for Gα(x2)Gα(x2) are necessary. Further for a general u>0u>0 we give an upper bound for the degree of factor of Gα(x)Gα(x) and Gα(x2)Gα(x2) in terms of u.