罗马诺夫斯基多项式、盖根鲍尔联系与su(1,1)梯级结构
Romanovski polynomials, Gegenbauer connections, and $\mathrm{su}(1,1)$ ladder structures
- Universidad Nacional de Educación a Distancia（西班牙国立远程教育大学）
- Universidad Autónoma de San Luis Potosí（圣路易斯波托西自治大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究探讨罗马诺夫斯基多项式的参数构造、与盖根鲍尔多项式的联系及su(1,1)梯级结构，推导相关递推关系与算子性质，为其模结构分析提供支撑。
AI中文摘要：
我们研究对应于依赖于次数的参数β_K=-K、α_K=c/(K+1)的首一罗马诺夫斯基（伪雅可比）多项式，其中K=n+ℓ。我们以显式形式写出在固定(α,β)下保持参数的一阶降阶与升阶关系，其比例常数为实数。将这些关系结合可得到任意超几何型族中存在的标准三项递推关系，迭代这些关系可从单位常数多项式出发，对R_n^{α,β}进行有序的一阶构造。我们还以有限三角形式求解了与α=0族的联系问题，将该族与盖根鲍尔多项式等同，并将梯级关系转移到(n,ℓ)格点。在代入x=cotχ后，能级内算子依赖于ℓ但不依赖于K，得到的修饰函数在ℓ固定的列上支持本征最低权su(1,1)模，而圆形行（n=0）需要显式边界规定与重标度。
英文摘要：
We study the monic Romanovski (pseudo-Jacobi) polynomials with degree dependent parameters $β_K=-K$ and $α_K=c/(K+1)$. We write out the parameter preserving first order ladder relations at fixed $(α,β)$, with real constants. Combining them returns the three-term recurrence; iterating them builds $R_n^{α,β}$ from the constant polynomial. We solve in triangular form the connection problem with the $α=0$ family, identified with the Gegenbauer polynomials, and transfer the ladder relations to the $(n,\ell)$ lattice. Under $x=\cotχ$ the in-level operators depend on $\ell$ alone. The dressed functions carry intrinsic lowest weight $\mathrm{su}(1,1)$ modules along columns of fixed $\ell$; rows need a boundary prescription and a rescaling. The deformed functions carry the same modules through higher order operators, which yield relations between Romanovski polynomials with different parameter pairs. Algebraically the family lies in the Jacobi class, and we indicate throughout which statements are inherited from that framework by transformation or specialization and which are not.
