论文标题
HAHN多项式和有理函数的统一代数基础
A unified algebraic underpinning for the Hahn polynomials and rational functions
论文作者
论文摘要
引入了三个发电机的代数表示$ M \ Mathfrak {h} $，并显示出允许Hahn代数和理性Hahn Algebra的嵌入。它具有变形的乔丹平面的真实版本，作为一个次级代理，其与Hahn多项式的联系得以建立。考虑与涉及发电机的特征值或广义特征值问题相对应的表示库。这些基础之间的重叠被证明是双光谱正交多项式或生物三相合理函数，从而基于$ M \ Mathfrak {h} $提供了对这些功能的统一描述。在差异和差异操作员方面，模型用于将基本特殊功能明确识别为HAHH多项式和有理功能，并确定其特征。出现了$ m \ mathfrak {h} $的嵌入在$ \ mathcal {u}中（\ mathfrak {sl} _2）$。获得二项式功能的PADé近似表作为副产品。
An algebra denoted $m\mathfrak{H}$ with three generators is introduced and shown to admit embeddings of the Hahn algebra and the rational Hahn algebra. It has a real version of the deformed Jordan plane as a subalgebra whose connection with Hahn polynomials is established. Representation bases corresponding to eigenvalue or generalized eigenvalue problems involving the generators are considered. Overlaps between these bases are shown to be bispectral orthogonal polynomials or biorthogonal rational functions thereby providing a unified description of these functions based on $m\mathfrak{H}$. Models in terms of differential and difference operators are used to identify explicitly the underlying special functions as Hahn polynomials and rational functions and to determine their characterizations. An embedding of $m\mathfrak{H}$ in $\mathcal{U}(\mathfrak{sl}_2)$ is presented. A Padé approximation table for the binomial function is obtained as a by-product.
