For a positive integer n and a real number α, the generalized Laguerre polynomials are defined by
These orthogonal polynomials are solutions to Laguerre's Differential Equation which arises in the treatment of the harmonic oscillator in quantum mechanics. Schur studied these Laguerre polynomials for its interesting algebraic properties. He obtained irreducibility results of and and derived that the Hermite polynomials H2n(x) and are irreducible for each n . In this article, we extend Schur's result by showing that the family of Laguerre polynomials and with , where d is the denominator of q, are irreducible for every n except when , n=2 where we give the complete factorization. In fact, we derive it from a more general result.
NGLC 2004-2010.National Geological Library of China All Rights Reserved.
Add:29 Xueyuan Rd,Haidian District,Beijing,PRC. Mail Add: 8324 mailbox 100083
For exchange or info please contact us via email.