Fixed-Lowering ${\mathfrak{sl}}_2$-Triples for Laurent-Shift Operators: Exact Stencil Endpoints and Recurrence Locality

Abstract

Let $S$ be the unit forward shift, and let $$\mathcal{A}=\mathbb{K}[x]⟨S,S^{-1}⟩/(Sx-(x+1)S)$$ 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 $X=x{S}^{-1}$, every completion is uniquely determined by $\lambda \in \mathbb{K}$ and $g\in \mathbb{K}[S,S^{-1}]$, with $$H=2(X+g)D-\lambda ,E=(X+g{)}^{2}D-\lambda (X+g).$$ 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 $\Delta$-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.

20 pp. under submitted to the "Journal of Difference Equations and Applications"