Extension of Laguerre polynomials with negative arguments II

For integers n,s,b0,…,bn with n≥3,s≥0, |b0|=|bn|=1, let G1(x)=G1(x,n,s)≔n!∑j=0nbj(j!)−1n+s−jn−jxj. For n≥0 and 0≤s≤92 it is proved in Shorey and Sinha (2022) that, except for finitely many pairs (n,s),G1(x)=G1(x,n,s) is either irreducible or linear factor times an irreducible polynomial. If s≤30, we...

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Bibliographic Details
Published in:Indagationes mathematicae 2022-07, Vol.33 (4), p.880-884
Main Authors: Shorey, T.N., Sinha, Sneh Bala
Format: Article
Language:English
Subjects:
Generalised Laguerre polynomials
Irreducibility
Polynomials
Primes
Valuations
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Summary:For integers n,s,b0,…,bn with n≥3,s≥0, |b0|=|bn|=1, let G1(x)=G1(x,n,s)≔n!∑j=0nbj(j!)−1n+s−jn−jxj. For n≥0 and 0≤s≤92 it is proved in Shorey and Sinha (2022) that, except for finitely many pairs (n,s),G1(x)=G1(x,n,s) is either irreducible or linear factor times an irreducible polynomial. If s≤30, we determine here explicitly the set of pairs (n,s) in the above assertion. This implies a new proof of the result of Nair and Shorey (2015) that G1(x) is irreducible for s≤22.
ISSN:0019-3577
1872-6100
DOI:10.1016/j.indag.2022.03.001