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On the Galois group of generalized Laguerre polynomials

Farshid Hajir (2005)

Journal de Théorie des Nombres de Bordeaux

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed α - < 0 , Filaseta and Lam have shown that the n th degree Generalized Laguerre Polynomial L n ( α ) ( x ) = j = 0 n n + α n - j ( - x ) j / j ! is irreducible for all large enough n . We use our criterion to show that, under these conditions, the Galois group of L n ( α ) ( x ) is either the alternating or symmetric group on n letters, generalizing results of Schur for α = 0 , 1 , ± 1 2 , - 1 - n .

On zeros of regular orthogonal polynomials on the unit circle

P. García Lázaro, F. Marcellán (1993)

Annales Polonici Mathematici

A new approach to the study of zeros of orthogonal polynomials with respect to an Hermitian and regular linear functional is presented. Some results concerning zeros of kernels are given.

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