On Galois groups of a one-parameter orthogonal family of polynomials

For a fixed integer t>1, we show that if t is not equal to 2, a square ≥4, or three times a square, then the discriminant of the generalized Laguerre polynomial L(s/t)n(x) is a nonzero square for at most finitely many pairs (n,s). Otherwise, the discriminant of L(s/t)n(x) is a nonzero square if and only if (n,s) belongs to one of seven explicitly describable infinite sets or to an additional finite set. This extends the results obtained for t=1 by P. Banerjee, M. Filaseta, C. Finch and J. Leidy. As a consequence, if α is a fixed rational number not equal to 1, 3, 5, or a negative integer, then for all but finitely many n, L(α)n(x) has Galois group Sn, thereby refining a previous result of M. Filaseta – T. Y. Lam and F. Hajir. As an illustration, we give for t=2 infinitely many integer specializations (n,s(n)) such that L(s(n)/2)n(x) has Galois group An. For n≤5, the set of rational numbers α for which the discriminant of L(α)n(x) is a nonzero square is explicitly computed by solving certain generalized Pell-like equations