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 (x) = Pn j=0 `n− j+r n− j ́ x / 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 andWong 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. Received by the editors September 5, 2006. This work was supported by the National Science Foundation under Grant No. 0226869. AMS subject classification: Primary: 11R09; secondary: 05E35. c ©Canadian Mathematical Society 2009. 583