Extension of Laguerre polynomials with negative arguments

• Published in: Indagationes mathematicae 2022-07, Vol.33 (4), p.801-815
• Main Authors: Shorey, T.N., Sinha, Sneh Bala
• Format: Article
• Language: English
• Subjects: Generalised Laguerre polynomials; Integers; Irreducibility; Polynomials; Primes; Valuations
• ISSN: 0019-3577; 1872-6100
• DOI: 10.1016/j.indag.2022.02.006

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)nLn(−n−s−1)(x)=∑j=0najxjj! and more general polynomial, let G(x)=∑j=0najbjxjj! 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 irreducible for 0≤s≤60. In this paper, by a different method, we prove: Apart from finitely many explicitly given possibilities, either G(x) is irreducible or G(x) is linear factor times irreducible polynomial. This is a consequence of the estimate s>1.9k 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.