2021/03/03 by T. N. Shorey, Shorey, T. N., Sneh Bala Sinha +1
Mathematics · #11A41 #11B25 #11C08 #11N05 #11N13 #11Z05 #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Identities #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2103.02353
openalex publication_date 2021/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the irreducibility of polynomial Ln(α) (x) where α is a negative integer. We observe that the constant term of Ln(α) (x) vanishes if and only if n ≥ |α| = -α. Therefore we assume that α= -n-s-1 where s is a non-negative integer. Let g(x) = (-1)n Ln(-n-s-1)(x) = ∑j=0n aj (xj)/(j!) and more general polynomial, let G(x) = ∑j=0n aj bj (xj)/(j!) where bj with 0 ≤ j ≤ n are integers such that |b0| = |bn| = 1. Schur was the first to prove the irreducibility of g(x) for s=0. It has been proved that g(x) is irreducibile for 0 ≤ s ≤ 60. In this paper, by a different method, we prove : Apart from finitely many explicitely given posibilities, either G(x) is irreducible or G(x) is linear factor times irreducible polynomial. This is a consequence of the estimate s > 1.9 k whenever G(x) has a factor of degree k ≥ 2 and (n,k,s) ≠ (10,5,4). This sharpens earlier estimates of Shorey and Tijdeman and Nair and Shorey.