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(−1−n−r)n(x)=∑nj=0(n−j+rn−j)xj/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.
CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES