论文标题
ASKEY方案中有限正交多项式的“ Diophantine”和分解特性
"Diophantine'' and Factorisation Properties of Finite Orthogonal Polynomials in the Askey Scheme
论文作者
论文摘要
探索了ASKEY方案中{\ em有限}正交多项式的``diophantine''的新解释和应用。相应的十二个多项式是（$ q $ - ）RACAH，（dual，$ q $ - ）Hahn，Krawtchouk和五种$ Q $ -Krawtchouk。这些（$ q $ - ）超几何多项式仅定义为$ 0,1，\ ldots，n $，构成$ n+1 $ 1 $二维三基因对称对称矩阵的特征向量的主要部分，与差异方程相对应。这些多项式的{\ em monic}版本都表现出``diophantine''的''''和分解属性，高度高于$ n $。这仅表示这些较高程度的多项式为$ n+1 $二维三二维真实对称矩阵的零norm``特征向量''。可以通过将较高程度的多项式作为相应矩阵特征值问题的多个darboux变换的种子解决方案来引入一种新型的属于这十二个多项式的新型多项式多项式。证明了最简单类型的多项式多项式的形状 - 不变特性。提出了显式转换公式。
A new interpretation and applications of the ``Diophantine'' and factorisation properties of {\em finite} orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the ($q$-)Racah, (dual, $q$-)Hahn, Krawtchouk and five types of $q$-Krawtchouk. These ($q$-)hypergeometric polynomials, defined only for the degrees of $0,1,\ldots,N$, constitute the main part of the eigenvectors of $N+1$-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The {\em monic} versions of these polynomials all exhibit the ``Diophantine'' and factorisation properties at higher degrees than $N$. This simply means that these higher degree polynomials are zero-norm ``eigenvectors'' of the $N+1$-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.
