Irreducibility and Galois groups of generalized Laguerre polynomials Ln(−1−n−r)(x

Abstract

We study the algebraic properties of Generalized Laguerre polynomials for negative integral values of a given parameter which is LFor different values of parameter r, this family provides polynomials which are of great interest. Hajir conjectured that for integers r ≥ 0 and n ≥ 1, L (−1−n−r) n (x) is an irreducible polynomial whose Galois group contains A n , the alternating group on n symbols. Extending earlier results of Schur, Hajir, Sell, Nair and Shorey, we confirm this conjecture for all r ≤ 60. We also prove that L (−1−n−r) n (x) is an irreducible polynomial whose Galois group contains A n whenever n > e r(1+ 1.2762 logr ) .

Extension of Laguerre polynomials with negative arguments

“…Like g (x), the results of Hajir on Newton polygons are not applicable for G(x) as the coefficients π j 's are not fixed. Therefore the proofs of our theorems are different from the proofs given in [6], [15] and [8]. Infact they follow the lines for the proofs of the results in [19].…”

“…Let 3|b 9 . Then the Newton polygons of G 1 (x) is a single edge joining (0, 0) to (10, 5) having lattice points (2, 1), (4, 2), (6,3) and (8,4). Therefore we may suppose that 3 ∤ b 9 .…”

“…Denote by S the set of all pairs (n, s) with 1 ≤ s ≤ 9 and (n, 2) ∈ T satisfying Lemma 2. Then S = {(9, 3), (9, 4), (9, 5), (9,6), (9, 7), (9,8), (9,9), (15,6), (15,7), (15,8), (15,9), (49, 7), (49, 8), (49, 9)}.…”

“…The irreducibilty of g(x) was proved first by Schur [20] for s = 0, by Hajir [5] for s = 1, by Sell [17] for s = 2 and by Hajir [6] for 3 ≤ s ≤ 8 and by Nair and Shorey [15] for 9 ≤ s ≤ 22. Further, developing on the work of Hajir [6] and Nair and Shorey [15], Jindal, Laishram and Sarma [8] extended the result of Nair and Shorey to 23 ≤ s ≤ 60. Moreover, Nair and Shorey [15] proved the following theorem.…”

“…The proof of results in [8] depends on Theorem 1. Further the weaker version of the above theorem was proved by Shorey and Tijdeman [19] for 0 ≤ s ≤ 0.95k.…”