Irreducibility of generalized Hermite-Laguerre polynomials III

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• 关键词：primary ; 11A41 ; 11B25 ; 11N05 ; 11N13 ; 11C08 ; 11Z05
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：164
• 期：Complete
• 页码：303-322
• 全文大小：449 K

文摘

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 $L_n^{(\pm \frac{1}{2})}(x)$ and $L_n^{(\pm \frac{1}{2})}(x^2)$ and derived that the Hermite polynomials H_2n(x)$H_{2n}(x)$ and $\frac{H_{2n+1}(x)}{x}$ are irreducible for each n  . In this article, we extend Schur's result by showing that the family of Laguerre polynomials $L_n^{(q)}(x)$ and $L_n^{(q)}(x^d)$ with $q\in \{\pm \frac{1}{3},\pm \frac{2}{3},\pm \frac{1}{4},\pm \frac{3}{4}\}$, where d is the denominator of q, are irreducible for every n   except when $q=\frac{1}{4}$, n=2$n=2$ where we give the complete factorization. In fact, we derive it from a more general result.