AI 中文总结
本研究分类了满足首一特征多项式有限项递推的微分算子$T=z^j\partial_z^j+z^m\partial_z^\ell$，给出递推存在的充要条件、闭式递推系数与差分算子阶，验证了Horozov--Shapiro--Tater的两个相关猜想。
AI 中文摘要
我们对微分算子$T=z^j\partial_z^j+z^m\partial_z^\ell$（其中$0\le m<\ell$且$1\le j<\ell$）进行分类，其首一特征多项式满足有限项递推关系。令$k=\ell-m$，此类递推存在当且仅当$j=1$且$k\mid\ell$。在该情形下，我们以闭式形式确定了所有递推系数，并证明相关差分算子的阶为$\ell$。我们还给出了显式因式分解，表明每个容许算子都属于Horozov--Shapiro--Tater猜想1.10中的第(2)类；微分阶与差分阶相等正是其猜想1.9所预测的结论。
英文摘要
We classify the differential operators \(T=z^j\partial_z^j+z^m\partial_z^\ell\), where \(0\le m<\ell\) and \(1\le j<\ell\), whose monic eigenpolynomials satisfy a finite-term recurrence relation. Writing \(k=\ell-m\), such a recurrence exists if and only if \(j=1\) and \(k\mid\ell\). In that case we determine all recurrence coefficients in closed form and prove that the associated difference operator has order \(\ell\). We also give an explicit factorization showing that every admissible operator is of Type~(2) in Conjecture~1.10 of Horozov--Shapiro--Tater; the equality of the differential and difference orders is the conclusion predicted by their Conjecture~1.9.
