Algebraic Properties of a Family of Generalized Laguerre Polynomials

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n >= 0, we conjecture that L(n)((-1-n-r)) (x) = Sigma(n)(j=0) (n-j) (n-j+r) x(j)/j! is a Q-irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r = n was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows from recent work of Hajir and Wong that the conjecture is true when r is large with respect to n >= 5. Here we verify it in three situations: (i) when n is large with respect to r, (ii) when r <= 8, and (iii) when n <= 4. The main tool is the theory of p-adic Newton Polygons.