Irreducibility of generalized Laguerre polynomials with integer u

详细信息    查看全文

• 作者：Shanta Laishram^a ; ^{shanta@isid.ac.in" class="auth_mail" title="E-mail the corresponding author} ; Saranya G. Nair^b ; ^{saranya@math.iitb.ac.in" class="auth_mail" title="E-mail the corresponding author} ; T.N. Shorey^b ; ^{shorey@math.iitb.ac.in" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 11A41 ; 11B25 ; 11N05 ; 11N13 ; 11C08 ; 11Z05
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：160
• 期：Complete
• 页码：76-107
• 全文大小：601 K

文摘

Generalized Laguerre polynomials $L_n^{(\alpha )}(x)$ are classical orthogonal polynomial sequences that play 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 α∈{0,1,−n−1}$\alpha \in \{0,1,-n-1\}$ are irreducible. For $\alpha =u+\frac{1}{2}$ with integer u   satisfying 1≤u≤45$1\le u\le 45$, 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)$(u,n)=(10,3)$ where we give a factorization. The cases u=−1,0$u=-1,0$ are due to Schur. Further we consider more general polynomials G_α(x)$G_{\alpha }(x)$ and G_α(x²)$G_{\alpha }(x^2)$ of degrees n and 2n  , respectively, and prove that they are either irreducible or have a factor of degree in {1,n−1}$\{1,n-1\}$, {1,2,2n−2,2n−1}$\{1,2,2n-2,2n-1\}$, respectively, except for an explicitly given finite set of pairs (u,n)$(u,n)$. We also show that these exceptional pairs other than one for G_α(x)$G_{\alpha }(x)$ and six for G_α(x²)$G_{\alpha }(x^2)$ are necessary. Further for a general u>0$u>0$ we give an upper bound for the degree of factor of G_α(x)$G_{\alpha }(x)$ and G_α(x²)$G_{\alpha }(x^2)$ in terms of u.