Algebraic properties of a family of Generalized Laguerre Polynomials

Original · EN

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r,n≥ 0, we conjecture that Lₙ⁽⁻¹⁻ⁿ⁻ʳ⁾(x) = ∑ⱼ₌₀ⁿ n-j+rn-jxʲ/j! is a -irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r=n was conjectured in the 50'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.