AI 中文总结
本文针对特征零域上有限洛朗移位算子代数，分类以固定下降算子$\boldsymbol{F=1-S}$的$\boldsymbol{\frak{sl}_2}$三元组，确定其模板端点与递推性质，刻画了正交性出现的条件。
AI 中文摘要
设$S$为单位前向移位算子，$$ \boldsymbol{\frak{A} = \big(\text{特征为零的域}\boldsymbol{\frak{K}}\text{上多项式系数的有限洛朗移位算子代数}\big) = \boldsymbol{\frak{K}[x]\big\rangle S,S^{-1}\big\big / \bigl(Sx-(x+1)S\bigr)} $$。我们对$\boldsymbol{\frak{A}}$中所有以固定下降算子$\boldsymbol{F=1-S}$的$\boldsymbol{\frak{sl}_2}$三元组进行分类。令$\boldsymbol{D=S-1}$，$\boldsymbol{X=xS^{-1}}$，每个完备三元组由$\boldsymbol{\frak{K}}$中的$\boldsymbol{\frak{\u03bb}}$和$\boldsymbol{\frak{K}[S,S^{-1}]}$中的$\boldsymbol{g}$唯一确定，其中$\boldsymbol{H=2(X+g)D-\u03bb}$，$\boldsymbol{E=(X+g)^2D-\u03bb(X+g)}$。我们给出从$\boldsymbol{H}$进行内在识别与重构的方法，确定$\boldsymbol{H}$和$\boldsymbol{E}$的精确极端移位，相关首一本征多项式构成$\boldsymbol{\u0394}$-Appell序列。当且仅当$\boldsymbol{g \u2208 S^{-1}\u039a[S]}$时，乘以$\boldsymbol{x}$具有有限低递推带宽，此时带宽等于Cartan模板的右端点；否则$\boldsymbol{H}$和$\boldsymbol{E}$仍为有限阶，而次数递推具有无限尾项，其最终带符号系数构成恢复$\boldsymbol{g}$最低洛朗项的多项式。在$\boldsymbol{\frak{R}}$上，正测度正交性恰好出现在平移首一Charlier系统中。
英文摘要
Let $S$ be the unit forward shift, and let $$ \mathcal{A} = \mathbb{K}[x]\langle S,S^{-1}\rangle \big/ \bigl(Sx-(x+1)S\bigr) $$ be the algebra of finite Laurent-shift operators with polynomial coefficients over a characteristic-zero field $\mathbb{K}$. We classify all $\mathfrak{sl}_2$-triples in $\mathcal{A}$ with fixed lowering operator $F=1-S$. Writing $D=S-1$ and $X=xS^{-1}$, every completion is uniquely determined by $λ\in\mathbb{K}$ and $g\in\mathbb{K}[S,S^{-1}]$, with $$ H=2(X+g)D-λ,\qquad E=(X+g)^2D-λ(X+g). $$ We give an intrinsic recognition and reconstruction from $H$ and determine the exact extreme shifts of $H$ and $E$. The associated monic eigenpolynomials form a $Δ$-Appell sequence. Multiplication by $x$ has finite lower recurrence bandwidth exactly when $g\in S^{-1}\mathbb{K}[S]$; in this case, the bandwidth equals the right endpoint of the Cartan stencil. Otherwise, $H$ and $E$ remain finite-order, while the degree recurrence has an infinite tail whose eventual signed coefficients form a polynomial recovering the lowest Laurent term of $g$. Over $\mathbb{R}$, positive-measure orthogonality occurs exactly for translated monic Charlier systems.
Comments20 pp. under submitted to the "Journal of Difference Equations and Applications"
