Romanovski polynomials, Gegenbauer connections, and su(1,1) ladder structures
Abstract
We study the monic Romanovski (pseudo-Jacobi) polynomials with degree dependent parameters βK=−K and αK=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 Rnα,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,ℓ) lattice. Under x=cotχ the in-level operators depend on ℓ alone. The dressed functions carry intrinsic lowest weight su(1,1) modules along columns of fixed ℓ; 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.
Related
- On zero distribution of the sum of Gegenbauer polynomials
- Permutation XOR cellular automata: partial-shift connections and half-primitive polynomials
- Homotopy Poisson and BV∞ structures on derived moduli spaces of flat connections
- Cycle structures and parities of linearized permutation polynomials
- Nonexistence of Strongly Regular Graphs via Gegenbauer Positivity and SRG Counting
BibTeX
@misc{vallejo2026romanovski,
title = {{Romanovski polynomials, Gegenbauer connections, and $\mathrmsu(1,1)$ ladder structures}},
author = {José A. Vallejo and Mariana Kirchbach},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.17221},
eprint = {2608.17221},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17221},
}
RIS
TY - UNPB
TI - Romanovski polynomials, Gegenbauer connections, and su(1,1) ladder structures
AU - José A. Vallejo
AU - Mariana Kirchbach
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2608.17221
UR - https://arxiv.org/abs/2608.17221
AB - 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,ℓ) lattice. Under x=cotχ the in-level operators depend on ℓ alone. The dressed functions carry intrinsic lowest weight su(1,1) modules along columns of fixed ℓ; 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.
ER -