An irreducibility question concerning modifications of Laguerre polynomials

Banerjee, Pradipto and Bera, Ranjan (2020) An irreducibility question concerning modifications of Laguerre polynomials. International Journal of Number Theory. pp. 1-21. ISSN 1793-0421

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Abstract

This paper addresses a question recently posed by Hajir concerning the irreducibility of certain modifications F(x) of generalized Laguerre polynomials L(−n−1−r)n(x) where r≥0 is an integer. For a fixed r≥0, we obtain lower bounds C(r) on n in terms of r such that F(x) is irreducible over the rationals for all n≥C(r). Furthermore, for r≤3, it is shown that F(x) is either irreducible or is a product of a linear polynomial and a polynomial of degree n−1. The set of circumstances in which F(x) has a linear factor for r≤3, is completely described.

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