Theoretical Statistics and Mathematics Unit, ISI Delhi

Irreducibility of Generalized Laguerre Polynomials $L_n^{(\frac{1}{2}+u)}(x)$ with integer $u$

by Shanta Laishram, Saranya G. Nair and T. N. Shorey

Generalized Laguerre polynomials $L_n^{(\alpha)}(x)$ are classical orthogonal polynomial sequences that plays an important role in various branches of analysis and mathematical physics. Schur (1929) was the first to study the algebraic properties of these polynomials by proving that $L_{n}^{(\alpha)}(x)$ where $\alpha \in \{0,1,-n-1\}$ are irreducible. For $\alpha=u+\frac{1}{2}$ with integer $u$ satisfying $1\leq u \leq 45$, we prove that $L_n^{(\alpha)}(x)$ and $L_n^{(\alpha)}(x^2)$ of degrees $n$ and $2n,$ respectively, are irreducible except when $(u,n)=(10,3)$ where we give a factorization. The cases $ u=-1,0$ are due to Schur. Further we consider more general polynomials $G_\alpha(x)$ and $G_\alpha(x^2)$
of degrees $n$ and $2n$, respectively, and prove that they are either irreducible or have a factor of degree in $\{1, n-1\}$, $\{1,2, 2n-2,2n-1\}$, respectively, except for an explicitly given finite set of pairs $(u, n)$. We also show that these exceptional pairs other than one for $G_{\alpha}(x)$ and six for $G_{\alpha}(x^2)$ are necessary. Further for a general $u >0$ we give an upper bound for the degree of factor of $G_{\alpha}(x)$ and $G_{\alpha}(x^2)$ in terms of $u.$

isid/ms/2015/14 [fulltext]

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