Algebraic Properties of a Family of Generalized Laguerre Polynomials

Abstract

Abstract. We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n ≥ 0, we conjecture thatis 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.

Irreducibility of Laguerre Polynomial Ln(−1−n−r)(x)Nair1, Shorey2 2015Indagationes Mathematicae83701View full textAdd to dashboardCite

“…In fact the method given by Coleman [2], further developed and refined by Filaseta [4], turns out to be very powerful for studying irreducibility of GLP. The case r = n gives Bessel polynomials and Filaseta and Trifonov [5] proved their irreducibility for all n. It has been proved by Hajir [6] and Sell [9] that L ⟨r ⟩ n (x) is irreducible and its Galois group contains A n when r = 1 or r = 2, respectively. More precisely, Hajir [6] proved that for n ≥ 14,…”

“…For an account of results obtained on GLP, we refer to Hajir [6] and Filaseta [1,4]. We shall restrict α to a negative integer in this paper.…”

“…Further Hajir [6] showed that for 3 ≤ r ≤ 8 and n ≥ 1, L ⟨r ⟩ n (x) is irreducible and G n (r ) contains A n . Also he proved that for n > B(r ) and r ≥ 9, L ⟨r ⟩ n (x) is irreducible and G n (r ) contains A n where…”

Harmonic divisors and rationality of zeros of Jacobi polynomials

“…It is still an unsolved question whether the Legendre polynomials are irreducible over the rationals, see [23], [24], [30], [40] and [41]. H. Ille has shown that P (0;0) n (x) has no quadratic factors which implies that P (0;0) n p b=c 6 = 0 for all n; b; c 2 N (even for the case b = 1; 3): In passing we note that recent research is devoted to the study of irreducibility of the Laguerre polynomials L n (x) initiated by I. Schur, see [20], [22], [36], and for a family of Jacobi polynomials see [12]. For general questions about irreducibility of polynomial with rational coe¢ cients we refer to [28], [31] and [38].…”

Irreducibility Criteria for Sums of Two Relatively Prime Polynomials

“…For applications of Newton's polygon method in the study of the irreducibility for various classes of polynomials, like for instance Bessel polynomials and Laguerre polynomials, we refer the reader to the work of Filaseta [9], [8] [15], [16], [17], and Sell [18].…”