Romanovski polynomials, Gegenbauer connections, and $\mathrm{su}(1,1)$ ladder structures

Abstract

We study the monic Romanovski (pseudo-Jacobi) polynomials with degree dependent parameters $\beta_K=-K$ and $\alpha_K=c/(K+1)$. We write out the parameter preserving first order ladder relations at fixed $(\alpha,\beta)$, with real constants. Combining them returns the three-term recurrence; iterating them builds $R_n^{\alpha,\beta}$ from the constant polynomial. We solve in triangular form the connection problem with the $\alpha=0$ family, identified with the Gegenbauer polynomials, and transfer the ladder relations to the $(n,\ell)$ lattice. Under $x=\cot\chi$ 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.