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On a certain family of generalized Laguerre polynomials. (English) Zbl 1053.11083

From the abstract: Following the work of Schur and Coleman, we prove the generalized Laguerre polynomial \(L_n^{(-3-n)}(x)=\sum_{j=0}^n \frac{1}{j!}\frac{(n-j+1)(n-j+2)}{2}x^j \) is irreducible over the rationals for all \(n\geq 1\) and has Galois group \(A_n\) if \(n+1\) is an odd square, and \(S_n\) otherwise. We also show that for certain negative integer values of \(\alpha\), Greek and certain congruence classes of \(n\) modulo 8, the splitting field of \(L_n^{\alpha}(x)\) can be embedded in a double cover.

MSC:

11R09 Polynomials (irreducibility, etc.)
11C08 Polynomials in number theory
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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