Hahn型正交性对于预解Delta算子 $\mathscr L=D(I-D)^{-1}$

中文摘要 AI 辅助

我们研究了在平移不变预解Delta算子 $\mathscr L=D(I-D)^{-1}$ 下的正交性。首先，我们证明了一个刚性定理：一个首一正交多项式序列是 $\mathscr L$-Appell 的当且仅当它是平移的 Laguerre 序列。然后我们超越 Appell 设定，证明了与 $$\mathcal S_\beta=\frac{\beta}{\beta+1}\mathcal L^0+\frac{1}{\beta+1}\delta$$ 相关的对角族满足精确的 Hahn 关系 $$\mathscr L B_{n+1}(x;\beta)=(n+1)B_n(x;\beta+1).$$ 第二个刚性结果表明，单个逐点关系 $P_{n+1}(c)=\theta\\,\mathscr L P_{n+1}(c)$ 已经强制正交性泛函为平移的 Laguerre 泛函或对角族的平移成员。连接和结构关系、半经典 Pearson 方程、微分刻画以及生成函数完成了这一描述。因此，预解算子提供了一个明确的设定，其中 Delta 算子降阶与半经典正交性通过精确的结构恒等式相联系。

英文摘要

We investigate orthogonality under the shift-invariant resolvent delta operator $\mathscr L=D(I-D)^{-1}$. First, we prove a rigidity theorem: a monic orthogonal polynomial sequence is $\mathscr L$-Appell if and only if it is a translated Laguerre sequence. We then move beyond the Appell setting and show that the diagonal family associated with $$\mathcal S_β=\fracβ{β+1}\mathcal L^0+\frac{1}{β+1}δ$$ satisfies the exact Hahn relation $$\mathscr L B_{n+1}(x;β)=(n+1)B_n(x;β+1).$$ A second rigidity result shows that a single pointwise relation $P_{n+1}(c)=θ\,\mathscr L P_{n+1}(c)$ already forces the orthogonality functional to be a translated Laguerre functional or a translated member of the diagonal family. Connection and structure relations, a semiclassical Pearson equation, a differential characterization, and a generating function complete the description. Thus the resolvent operator provides an explicit setting in which delta-operator lowering and semiclassical orthogonality are linked by exact structural identities.

发表机构

• Le Mans Université（勒芒大学）
• University of Gabes, Faculty of Sciences of Gabes（加贝斯大学，加贝斯理学院）

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