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On the genus of generalized Laguerre polynomials. (English) Zbl 1078.33009

Recently Hajir and Wong [unpublish paper mentioned in the reference] have proved that for \(n\geq 5\) and for any number field \(K\), the Galois group of \(L_n^{(\alpha)}(x)\) [generalised Laguerre polynomials] over \(K\) in \(S_n\) for all but finitely many \(\alpha\in K\). In this proof the main thing is to compute the genus of the function field in the splitting field of \(L_n^{(t)}(x)\) over the function field \(Q(t)\). The authors have proved that for \(n\geq 5\) \(L_n^{(t)}(x)\) defines an absolutely irreducible plane curve of geometric genus \(>1\). They determined the exact genus of these curves and also claimed that their technique should be applicable to other families of orthogonal polynomials that satisfy recursion similar to those for \(L_n^{(t)}(x)\).

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C90 Applications of hypergeometric functions
33E99 Other special functions
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References:

[1] Feit, W., \( \widetilde{A}_5\) and \(\widetilde{A}_7\) are Galois groups over number fields, J. Algebra, 104, 231-260 (1986)
[2] Gow, R., Some generalized Laguerre polynomials whose Galois groups are the alternating groups, J. Number Theory, 31, 201-207 (1989) · Zbl 0693.12009
[3] Hajir, F., Some \(\widetilde{A}_n\)-extensions obtained from generalized Laguerre polynomials, J. Number Theory, 50, 206-212 (1995) · Zbl 0829.12004
[4] F. Hajir, S. Wong, Specializations of oneparameter families of polynomials, preprint, 2004; F. Hajir, S. Wong, Specializations of oneparameter families of polynomials, preprint, 2004
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